Polynomial progressions in topological fields

Abstract

Let $P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$ be polynomials with distinct degrees, no constant terms and coefficients in a general local field $\mathbb {K}$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots , x + P_m(y)$ lying in a set $S\subseteq \mathbb {K}$ of positive density. The proof relies on a general $L^{\infty }$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and p-adic analysis.

1 Introduction

Szemerédi’s famous theorem [43] states that any set S of integers with positive (upper) density must necessarily contain arbitrarily long arithmetic progressions. Quantitative versions have been obtained by several authors, first by Roth [40] for three-term arithmetic progressions and by Gowers [18] in general, with the current best bounds due to Bloom and Sisask [7], Kelley and Meke [21] in the three-term case and Leng, Sah and Sawhney [26] for longer progressions (see also Green and Tao [17] and Gowers [18]). More generally, one can consider polynomial progressions $x, x+P_1(y), \ldots , x + P_m(y)$ for $x,y \in {\mathbb Z}$ with $y\not = 0$ , where $P_j \in {\mathbb Z}[\mathrm {y}]$ is a sequence of polynomials with integer coefficients and no constant terms (the case of arithmetic progressions corresponding to linear polynomials). Bergelson and Leibman [6], extending earlier work of Bergelson, Furstenberg and Weiss [5], generalised Szemerédi’s theorem to polynomial progressions. Obtaining quantitative versions of Bergelson and Leibman’s result has been a challenging problem and no progress (outside a few results on two-term progressions) has been made until very recently.

Inspired by the earlier work of Bergelson, Furstenberg and Weiss, Bourgain obtained a quantitative lower bound on the count of three-term polynomial progressions in the setting of the real field ${\mathbb R}$ . He accomplished this by coupling a technique he developed in his work on arithmetic progressions [2], together with Fourier-analytic methods.

Theorem 1.1 (Bourgain [3]).

Given $\varepsilon>0$ , there exists a $\delta (\varepsilon )>0$ such that for any $N\ge 1$ and measurable set $S \subseteq [0,N]$ satisfying $|S\cap [0,N]|\ge \varepsilon N$ , we have

(1.2) $$ \begin{align} \bigl|\{(x,y) \in [0,N]\times[0,N^{1/d}] : x, x+y, x + y^d \in S \}\bigr| \ \ge \ \delta N^{1+1/d}. \end{align} $$

In particular, we have the existence of a triple $x, x+y$ and $x+y^d$ belonging to S with y satisfying the gap condition $y \ge \delta N^{1/d}$ .

The bound (1.2) implies a quantitative multiple recurrence result. Only recently have there been extensions to more general three-term progressions $x, x +P_1(y), x + P_2(y)$ ; see the work of Durcik, Guo and Roos [11] when $P_1(y) = y$ and general $P_2$ and of Chen, Guo and Li [8] for general $P_1, P_2 \in {\mathbb R}[\mathrm {y}]$ with distinct degrees. The methods in these papers, using delicate oscillatory integral operator bounds, seem limited to three-term progressions.

In another direction, Bourgain and Chang [4] gave quantitative bounds for three-term progressions of the form $x, x+y, x+y^2$ in the setting of finite fields ${\mathbb F}_q$ . This result was extended to more general three-term polynomial progressions by Peluse [36] and Dong, Li and Sawin [10]. The techinques in these papers, using a Fourier-analytic approach which relies on sophisticated exponential sum bounds over finite fields, also seem limited to three-term progressions.

By using new ideas in additive combinatorics, by-passing the need of inverse theorems for Gowers’ uniformity norms of degree greater than 2, Peluse [37] recently made a significant advance, giving quantitative bounds for general polynomial progressions $x, x+P_1(y), \ldots , x+P_m(y)$ in ${\mathbb F}_q$ , where $\{P_1,\ldots , P_m\} \subseteq {\mathbb Z}[\mathrm {y}]$ are linearly independent over ${\mathbb Q}$ .

Inspired by this work, Peluse and Prendiville [39] obtained the first quantitative bounds for three-term polynomal progressions in the setting of the integers ${\mathbb Z}$ . This has been extended recently to general polynomial progressions $x, x+P_1(y), \ldots , x + P_m(y)$ with $P_j \in {\mathbb Z}[\mathrm {y}] $ having distinct degrees by Peluse [38]. So although the first quantitative bounds for polynomial progressions were made in the setting of the real field ${\mathbb R}$ , we have seen major advances in both the finite field ${\mathbb F}_q$ and integer ${\mathbb Z}$ settings by employing new ideas in additive combinatorics.

One purpose of this paper is to rectify this situation for the continuous setting by establishing quantitative bounds for general polynomial progressions in the real field ${\mathbb R}$ , bringing it in line with the recent advances in the finite field and integer settings. Another purpose is to illustrate how one can marry these new ideas in additive combinatorics with other ideas, notably from the work of Krause, Mirek and Tao [23] to obtain compactness results for general multilinear polynomial averaging operators which have implications for problems in euclidean harmonic analysis. These ideas and arguments are robust enough to allow us to obtain quantitative bounds for polynomial progressions in a general local field.

Theorem 1.3. Let $\mathbb {K}$ be a local field with Haar measure $\mu $ . Let ${\mathcal P} = \{P_1, \ldots , P_m\}$ be a sequence of polynomials in $\mathbb {K}[\mathrm {y}]$ with distinct degrees and no constant terms, and let d denote the largest degree among the polynomials in ${\mathcal P}$ . When $\mathbb {K}$ has positive characteristic, we assume the characteristic is larger than d.

For any $\varepsilon>0$ , there exists a $\delta (\varepsilon , {\mathcal P})> 0$ and $N(\varepsilon , {\mathcal P}) \ge 1$ such that for any $N \ge N(\varepsilon , {\mathcal P})$ and measurable set $S \subseteq \mathbb {K}$ satisfying $\mu (S \cap B_N) \ge \varepsilon N$ , we have

(1.4) $$ \begin{align} \mu\bigl(\{(x,y) \in B_N \times B_{N^{1/d}} : x, x+P_1(y),\ldots x + P_m(y) \in S \}\bigr) \ \ge \ \delta N^{1+1/d}. \end{align} $$

In particular, we have the existence of a progression $x, x+P_1(y), \ldots , x+ P_m(y)$ belonging to S with y satisfying the gap condition $|y| \ge \delta N^{1/d}$ . The proof will show that we can take $\delta = \varepsilon ^{C\varepsilon ^{-2m-2}}$ for some $C = C_{\mathcal P}> 0$ and $N(\varepsilon , {\mathcal P}) = \varepsilon ^{-C' \varepsilon ^{-2m-2}}$ for a slightly larger $C'> C_{\mathcal P}$ .

When $\mathbb {K} = {\mathbb R}$ is the real field, Theorem 1.3 extends the work in [3], [11] and [8] from three-term polynomial progressions to general polynomial progressions albeit for large N, depending on $\varepsilon $ .

When $\mathbb {K}={\mathbb C}$ , Theorem 1.3 represents the first known results for complex polynomial progressions. The absolute value $|\cdot |$ used in the statement of Theorem 1.3 is normalised so that we can express the result in this generality (see Section 3). For any sequence of complex polynomials $\{P_1,\ldots , P_m\} \subseteq {\mathbb C}[\mathrm {z}]$ with distinct degrees and $P_j(0) = 0$ , Theorem 1.3 has the following consequence: Given $\varepsilon>0$ , there is a $\delta>0$ such that for sufficiently large N and any set S in the complex plane satisfying $|S\cap {\mathbb D}_N| \ge \varepsilon N^2$ , we can find a progression of the form $w, w+ P_1(z), \ldots , w + P_m(z)$ lying in S such that $|z| \ge \delta N^{2/d}$ .

Important in our analysis are certain properties for $m+1$ linear forms formed from our collection ${\mathcal P} = \{P_1, \ldots , P_m\} \subseteq \mathbb {K}[\mathrm {y}]$ of m polynomials with distinct degrees, say $1\le \deg (P_1) < \ldots < \deg (P_m) =: d$ . Let $N \ge 1$ and consider the form

$$ \begin{align*}\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m):= \frac{1}{N^d} \int_{\mathbb{K}^2}f_0(x)\prod_{i=1}^mf_{i}(x-P_i(y))d\mu_{[N]}(y)d\mu(x). \end{align*} $$

Here, is normalised measure on the ball $B_N(0)$ (we will describe notation used in the paper in Section 4). The key result in the proof of Theorem 1.3 is the following $L^{\infty }$ inverse theorem for $\Lambda _{{\mathcal P}; N}$ which is of independent interest.

Theorem 1.5 (Inverse theorem for $(m+1)$ -linear forms).

With the setup above, let $f_0, f_1,\ldots , f_m$ be $1$ -bounded functions supported on a ball $B\subset \mathbb {K}$ of measure $N^d$ . Suppose that

$$ \begin{align*} |\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m)|\ge\delta. \end{align*} $$

Then there exists $N_1\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_1)}$ such that

$$ \begin{align*} N^{-d}\big\| \mu_{[N_1]}*f_1\big\|_{L^1(\mathbb{K})} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}. \end{align*} $$

The main application of Theorem 1.5 for us will be to prove a precise structural result for multilinear polynomial operators of the form

$$ \begin{align*} A_N^{\mathcal P}(f_1,\ldots, f_m)(x) \ = \ \int_{\mathbb{K}} f_1(x + P_1(y)) \cdots f_m(x + P_m(y)) \, d\mu_{[N]}(y). \end{align*} $$

We will use ideas in the recent work of Krause, Mirek and Tao [23] to accomplish this, and consequently, we will be able to establish the following important Sobolev estimate.

Theorem 1.6 (A Sobolev inequality for $A_N^{\mathcal P}$ ).

Let $1<p_1,\ldots , p_m<\infty $ satisfying $\frac {1}{p_1}+\ldots +\frac {1}{p_m}=1$ be given. Then for $N_j\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_j)}$ , we have

$$ \begin{align*} \|A_N^{\mathcal P}(f_1,\ldots,f_{j-1}, (\delta_0-\varphi_{N_j})*f_j,f_{j+1}\ldots, f_m)\|_{L^1(\mathbb{K})} \lesssim \delta^{1/8} \prod_{i=1}^{m} \|f_i\|_{L^{p_i}(\mathbb{K})}, \end{align*} $$

provided $N\gtrsim \delta ^{-O_{\mathcal P}(1)}$ . Here, $\varphi _{N_j}$ is a smooth cut-off function such that ${\widehat {\varphi _{N_j}}}(\xi ) \equiv 1$ for $\xi \in B_{{N_j}^{-1}}(0)$ .

Following an argument of Bourgain in [3], we will show how Theorem 1.6 implies Theorem 1.3. Versions of Theorem 1.6 for two real polynomials $\{P_1, P_2\} \subseteq {\mathbb R}[\mathrm {y}]$ were established in [3], [11] and [8] using delicate oscillatory integral operator bounds. Our arguments are much more elementary in nature and do not require deep oscillatory integral/exponential sum/character sum bounds outside a standard application of van der Corput bounds (see [41]) when $\mathbb {K} = {\mathbb R}$ or Hua’s exponential sum bound [13] when $\mathbb {K} = {\mathbb Q}_p$ (which extends Mordell’s classical bound from the finite field setting to complete exponenial sums over ${\mathbb Z}/p^m{\mathbb Z}$ ) – these bounds extend readily to any local field ${\mathbb K}$ ; see Section 3. Furthermore, the Sobolev inequalities in [11] and [8] were only established for certain sparse sequences of scales N. The bound in Theorem 1.6 holds for all sufficiently large scales N.

The Sobolev bound in Theorem 1.6 potentially has many other applications. See [3] for a discussion on the implications of Theorem 1.6 to compactness properties of the multilinear operator $A_N^{\mathcal P}$ . Pointwise convergence results for multilinear polynomial averages are common applications of such Sobolev bounds. See [8] where the Sobolev inequality is used to prove the existence of polynomial progressions in sets of sufficiently large Hausdorff dimension. See also [22], [24], [19], [20] and [9].

Our results require the scales N to be large. It would be interesting, for various applications, to establish these results for small scales as well.

2 Structure of the paper

After a review of analysis in the setting of local fields, including some essential but basic oscillatory integral bounds, we set up some notation and detail some tools involving the Gowers uniformity norms. In Section 5, we give some preliminary results necessary to carry out the core arguments. In Section 6, we give the proof of Theorem 1.5 which is based on a polynomial ergodic theorem (PET) induction scheme and a degree lowering argument developed by the third author in earlier work. In Section 7, we will prove Theorem 1.6. Finally, in Section 8, we show how Theorem 1.3 follows as a consequence of Theorem 1.6.

3 Review of basic analysis on local fields

A basic reference for the material reviewed in this section is [35].

Let $\mathbb {K}$ be a locally compact topological field with a nondiscrete topology. Such fields are called local fields and have a unique (up to a positive multiple) Haar measure $\mu $ . They also carry a nontrivial absolute value $|\cdot |$ such that the corresponding balls $B_r(x) = \{y \in \mathbb {K}: |y-x|\le r\}$ generate the topology.

Recall that an absolute value on a field $\mathbb {K}$ is a map $|\cdot | : \mathbb {K} \to {\mathbb R}^{+}$ satisfying

$$ \begin{align*} (a) \ |x| = 0 \ \Leftrightarrow \ x = 0, \ \ \ (b) \ |xy| = |x||y| \ \ \ \mathrm{and} \ \ (c) \ |x+y| \le C (|x| + |y|) \end{align*} $$

for some $C\ge 1$ . It is nontrivial if there is an $x\not = 0$ such that $|x| \not = 1$ . Two absolute values $|\cdot |_1$ and $|\cdot |_2$ are said to be equivalent if there is a $\theta> 0$ such that $|x|_2 = |x|_1^{\theta }$ for all $x\in \mathbb {K}$ . Equivalent absolute values give the same topology. There is always an equivalent absolute value such that the triangle inequality $(c)$ holds with $C = 1$ . If $|\cdot |$ satisfies the stronger triangle inequality $(c') \, |x+y| \le \max (|x|,|y|)$ , we say that $|\cdot |$ is non-Archimedean. Note that if $|\cdot |$ is non-Archimedean, then all equivalent absolute values are non-Archimedean. The field $\mathbb {K}$ is said to be non-Archimedean if the underlying absolute value (and hence all equivalent ones) is non-Archimedean. Otherwise, we say $\mathbb {K}$ is Archimedean.

When $\mathbb {K}$ is Archimedean, then it is isomorphic to the real ${\mathbb R}$ or complex ${\mathbb C}$ field with the usual topology. In this case, Haar measure is a multiple of Lebesgue measure. When $\mathbb {K}$ is non-Archimedean, then it is a finite extension of a p-adic field ${\mathbb Q}_p$ in the characteristic zero case and a function field of Laurent series over a finite field in the positive characteristic case. Furthermore, the ring of integers $o_{\mathbb {K}} := \{x \in \mathbb {K}: |x|\le 1\}$ and the unique maximal ideal $m_{\mathbb {K}} := \{x \in \mathbb {K} : |x| < 1\}$ do not depend on the choice of absolute value (it is invariant when we pass to an equivalent absolute value). For any $\mathbb {K}$ , we normalise Haar measure so that $\mu (B_1(0)) = 1$ .

When $\mathbb {K}$ is non-Archimedean, the unique maximal ideal $m_{\mathbb {K}} = (\pi )$ is principal and we call any generating element $\pi $ a uniformizer. Furthermore, the residue field $k := o_{\mathbb {K}}/m_{\mathbb {K}}$ is finite, say with q elements. For $x\in \mathbb {K}$ , there is a unique $n\in {\mathbb Z}$ such that $x = \pi ^n u$ where u is a unit. We can go further and expand any $x\in \mathbb {K}$ as a Laurent series in $\pi $ ; $x = \sum _{j\ge -L} x_j \pi ^j$ , where each $x_j$ belongs to the residue field k. If $x_{-L} \not =0$ , then $x = \pi ^{-L} u$ , where $u = \sum _{j\ge -L} x_j \pi ^{j+L}$ is a unit.

There is a choice of (equivalent) absolute value $|\cdot |$ such that $\mu (B_r(x)) \simeq r$ for all $r>0$ and $x\in \mathbb {K}$ . When $\mathbb {K}= {\mathbb R}$ , we have $|x| = x \, \mathrm {sgn}(x)$ and when $\mathbb {K} = {\mathbb C}$ , we have $|z| = z{\overline {z}}$ . When $\mathbb {K}$ is non-Archimedean, then the absolute value $|x| := q^{-m}$ , where $x = \pi ^m u$ and u a unit has the property that its balls satisfy $\mu (B_r(x)) = q^n$ , where $q^n \le r < q^{n+1}$ and so $\mu (B_r(x)) \simeq r$ . We choose the absolute value with this normalisation.

We will need a couple simple change of variable formulae which we will use again and again:

$$ \begin{align*} \int_{\mathbb{K}} f(x + y) \, d\mu(x) \ = \ \int_{\mathbb{K}} f(x) \, d\mu(x) \ \ \mathrm{and} \ \ \int_{\mathbb{K}} f(y^{-1} x) \, d\mu(x) \ = \ |y| \, \int_{\mathbb{K}} f(x) \, d\mu(x). \end{align*} $$

The first follows from the translation invariance of the Haar measure $\mu $ . For the second formula, the measure $E \to \mu (yE)$ defined by an element $y\in \mathbb {K}$ is translation-invariant and so by the uniqueness of Haar measure, we have $\mu (yE) = \mathrm {mod}_{\mu }(y) \mu (E)$ for some nonnegative number $\mathrm {mod}_{\mu }(y)$ , the so-called modulus of the measure $\mu $ . In fact $|y| := \mathrm {mod}_{\mu }(y)$ defines the absolute value with the desired normalisation whose balls $B_r(x)$ satisfy $\mu (B_r(x)) \simeq r$ . This proves the second change of variables formula. There is one additional, more sophisticated, nonlinear change of variable formula which we will need at one point, but we will justify this change of variables at the time.

The (additive) character group of $\mathbb {K}$ is isomorphic to itself. Starting with any nonprincipal character $\mathrm {e}$ on $\mathbb {K}$ , all other characters $\chi $ can be identified with an element $y\in \mathbb {K}$ via $\chi (x) = \mathrm {e}(yx)$ . We fix a convenient choice for $\mathrm {e}$ ; when $\mathbb {K}={\mathbb R}$ , we take $\mathrm {e}(x) = e^{2\pi i x}$ . When $\mathbb {K}$ is non-Archimedean, we choose $\mathrm {e}$ so that $\mathrm {e} \equiv 1$ on $o_{\mathbb {K}}$ and nontrivial on $B_q(0)$ ; that is, there is a $x_0$ with $|x_0| = q$ such that $\mathrm {e}(x_0) \not = 1$ . The choice of $\mathrm {e}$ on ${\mathbb C}$ does not really matter but a convenient choice is $\mathrm {e}(z) = e^{2\pi i \operatorname {\mathrm {Re}}{z}}$ . We define the Fourier transform

$$ \begin{align*} {\widehat{f}}(\xi) \ = \ \int_{\mathbb{K}} f(x) \mathrm{e}(-\xi x) \, d\mu(x). \end{align*} $$

Plancherel’s theorem and the Fourier inversion formula hold as in the real setting.

3.1 An oscillatory integral estimate

For $P(x) = a_d x^d + \cdots + a_1 x \in \mathbb {K}[\mathrm {x}]$ , we will use the following oscillatory integral bound:

(3.1) $$ \begin{align} |I(P)| \ \le \ C_d \, [\max_j |a_j| ]^{-1/d} \ \ \ \mathrm{where} \ \ \ I(P) \ = \ \int_{B_1(0)} \mathrm{e}(P(x)) \, d\mu(x). \end{align} $$

When $\mathbb {K} = {\mathbb R}$ , it is a simple matter to deduce the bound (3.1) from general oscillatory bounds due to van der Corput (see [41]). When $\mathbb {K} = {\mathbb Q}_p$ is the p-adic field, then

$$ \begin{align*}I(P) \ = \ p^{-s} \sum_{x=0}^{p^s -1} e^{2\pi i Q(x)/p^s} \ \ \ \mathrm{where} \ \ \ p^s = \max_j |a_j| \ \ \mathrm{and} \ \ Q(x) = b_d x^d + \cdots + b_1 x \in {\mathbb Z}[\mathrm{x}] \end{align*} $$

satisfies $\mathrm {gcd}(b_d, \ldots , b_1, p) = 1$ ; hence, a classical result of Hua [13] implies $|I(P)| \le C_d p^{-s/d}$ which is Equation (3.1) in this case. It is natural to extend Hua’s bound to other non-Archimedean fields; see, for example, [45] where character sums are treated over general Dedekind domains which in particular establishes Equation (3.1) for any non-Archimedean field $\mathbb {K}$ when the characteristic of $\mathbb {K}$ (if positive) is larger than d, a basic assumption appearing in our main result Theorem 1.3.

It is not straightforward to apply van der Corput bounds when $\mathbb {K} = {\mathbb C}$ . However, we can see the bound (3.1) for both $\mathbb {K} = {\mathbb R}$ and $\mathbb {K} = {\mathbb C}$ as a consequence of the following general bound due to Arkhipov, Chubarikov and Karatsuba [1]: Let $P \in {\mathbb R}[X_1, \ldots , X_n]$ be a real polynomial of degree d in n variables. If ${\mathbb B}^n$ denotes the unit ball in ${\mathbb R}^n$ , then

(3.2) $$ \begin{align} \Big| \int_{{\mathbb B}^n} e^{2\pi i P({\underline{x}})} \, d{\underline{x}} \Big| \ \le \ C_{d,n} \, H(P)^{-1} \ \ \ \mathrm{where} \ \ \ H(P) = \min_{{\underline{x}}\in {\mathbb B}^n} \max_{\alpha} |\partial^{\alpha} P({\underline x})|^{1/|\alpha|}. \end{align} $$

A simple equivalence of norms argument shows that $H(P) \ge c_d [\max _{\alpha } |a_{\alpha }|]^{1/d}$ , where $P({\underline {x}}) = \sum _{\alpha } a_{\alpha } {\underline {x}}^{\alpha }$ and d is the degree of P. Hence, Equation (3.2) implies Equation (3.1) when $\mathbb {K} = {\mathbb R}$ . When $\mathbb {K} = {\mathbb C}$ and $f(z) = a_d z^d + \cdots + a_1 z \in {\mathbb C}[\mathrm {z}]$ , write $f(x+iy) = P(x,y) + i Q(x,y)$ and note that

$$ \begin{align*}\int_{B_1(0)} \mathrm{e}(f(z)) \, dz \ = \ \int_{{\mathbb B}^2} e^{2\pi i P(x,y)} \, dx dy \end{align*} $$

for the choice of character $\mathrm {e}(z) = e^{2\pi i \operatorname {\mathrm {Re}}{z}}$ . From the Cauchy–Riemann equations, we have $H(P) \simeq _d \min _{|z|\le 1} \max _k |f^{(k)}(z)|^{1/2k} \ge c_d [\max _j |a_j|]^{1/2d}$ (recall we are using the absolute value $|z| = z{\overline {z}}$ on ${\mathbb C}$ ), and so Equation (3.2) implies Equation (3.1) with exponent $1/2d$ in this case. There is an alternative argument which establishes Equation (3.1) with the exponent $1/d$ when $\mathbb {K} = {\mathbb C}$ but this is unimportant for our purposes.

4 Some notation and basic tools

By a scale N, we mean a positive number when $\mathbb {K}$ is Archimedean and when $\mathbb {K}$ is non-Archimedean, it denotes a discrete value $N = q^k, \, k\in {\mathbb Z}$ , a power of the cardinality of the residue field k. When N is a scale, we denote by $[N] := B_N(0)$ the ball with centre $0$ and radius N. In this case, we have $\mu ([N]) \simeq N$ (equality in the non-Archimedean case) by our normalisations of the absolute value $|\cdot |$ and Haar measure $\mu $ . An interval I is a ball $I = B_{r_I}(x_I)$ with some centre $x_I \in \mathbb {K}$ and radius $r_I>0$ . For an interval I, we associate the measure

For an interval I, we define the Fejér kernel and the corresponding measure $d\nu _I(x) = \kappa _I(x) d\mu (x)$ . When $I = [N]$ for some scale N, we have $-I = I$ and so . Furthermore, when $\mathbb {K}$ is non-Archimedean, we have and so $d\nu _I = d\mu _I$ in this case. When $\mathbb {K} = {\mathbb R}$ and $I = [0,N]$ , we have $\kappa _I(x) = N^{-1}(1 - |x|/N)$ when $|x|\le N$ and zero otherwise.

We now give precise notation which we will use throughout the paper.

4.1 Basic notation

As usual, $\mathbb Z$ will denote the ring of rational integers.

1. 1. We use $\mathbb Z_+:=\{1, 2,\ldots \}$ and $\mathbb {N} := \mathbb Z_+\cup \{0\}$ to denote the sets of positive integers and nonnegative integers, respectively.

2. 2. For any $L\in \mathbb R_+$ , we will use the notation

   

3. 3. We use to denote the indicator function of a set A. If S is a statement, we write to denote its indicator, equal to $1$ if S is true and $0$ if S is false. For instance, .

4.2 Asymptotic notation and magnitudes

The letters $C,c, C_0, C_1, \ldots>0$ will always denote absolute constants; however, their values may vary from occurrence to occurrence.

1. 1. For two nonnegative quantities $A, B$ , we write $A \lesssim _{\delta } B$ ( $A \gtrsim _{\delta } B$ ) if there is an absolute constant $C_{\delta }>0$ (which possibly depends on $\delta>0$ ) such that $A\le C_{\delta }B$ ( $A\ge C_{\delta }B$ ). We will write $A \simeq _{\delta } B$ when $A \lesssim _{\delta } B$ and $A\gtrsim _{\delta } B$ hold simultaneously. We will omit the subscript $\delta $ if irrelevant.

2. 2. For a function $f:X\to \mathbb C$ and positive-valued function $g:X\to (0, \infty )$ , write $f = O(g)$ if there exists a constant $C>0$ such that $|f(x)| \le C g(x)$ for all $x\in X$ . We will also write $f = O_{\delta }(g)$ if the implicit constant depends on $\delta $ . For two functions $f, g:X\to \mathbb C$ such that $g(x)\neq 0$ for all $x\in X$ we write $f = o(g)$ if $\lim _{x\to \infty }f(x)/g(x)=0$ .

4.3 Polynomials

Let $\mathbb {K}[\mathrm {t}]$ denote the space of all polynomials in one indeterminate $\mathrm {t}$ with coefficients in $\mathbb {K}$ . Every polynomial $P\in \mathbb {K}[\mathrm {t}]$ can be written as a formal power series

(4.1) $$ \begin{align} P(t)=\sum_{j=0}^{\infty}c_jt^j, \end{align} $$

where all but finitely many coefficients $c_j\in \mathbb {K}$ vanish.

1. 1. We define the degree of $P\in \mathbb {K}[\mathrm {t}]$ by

   $$ \begin{align*} \deg(P):=&\max\{j\in {\mathbb Z}_+: c_j\neq0\}. \end{align*} $$

2. 2. A finite collection ${\mathcal P}\subset \mathbb {K}[\mathrm {t}]$ has degree $d\in \mathbb {N}$ , if $d=\max \{\deg (P): P\in {\mathcal P}\}$ .

3. 3. For a polynomial $P\in \mathbb {K}[\mathrm {t}]$ and $j\in \mathbb {N}$ , let $\operatorname {\mathrm {c}}_j(P)$ denote j-th coefficient of P. We also let $\ell (P)$ denote the leading coefficient of P; that is, for P as in Equation (4.1) we have $\operatorname {\mathrm {c}}_j(P)=c_j$ for $j\in \mathbb {N}$ and $\ell (P)=c_{d}$ where $d = \deg {P}$ .

4.4 $L^p$ spaces

$(X, {\mathcal B}(X), \lambda )$ denotes a measure space X with $\sigma $ -algebra ${\mathcal B}(X)$ and $\sigma $ -finite measure $\lambda $ .

1. 1. The set of $\lambda $ -measurable complex-valued functions defined on X will be denoted by $L^0(X)$ .

2. 2. The set of functions in $L^0(X)$ whose modulus is integrable with p-th power is denoted by $L^p(X)$ for $p\in (0, \infty )$ , whereas $L^{\infty }(X)$ denotes the space of all essentially bounded functions in $L^0(X)$ .

3. 3. We will say that a function $f\in L^0(X)$ is $1$ -bounded if $f\in L^{\infty }(X)$ and $\|f\|_{L^{\infty }(X)}\le 1$ .

4. 4. For any $n\in \mathbb Z_+$ the measure $\lambda ^{\otimes n}$ will denote the product measure $\lambda \otimes \ldots \otimes \lambda $ on the product space $X^n$ with the product $\sigma $ -algebra ${\mathcal B}(X)\otimes \ldots \otimes {\mathcal B}(X)$ .

4.5 Gowers box and uniformity norms

We will use the Gowers norm and Gowers box norm of a function f which is defined in terms of the multiplicative discrete derivatives $\Delta _{h_1.\ldots , h_s} f(x)$ : for $x, h \in \mathbb {K}$ , we set $\Delta _h f(x) = f(x){\overline {f(x+h)}}$ , and iteratively, we define

$$ \begin{align*} \Delta_{h_1, \ldots, h_s} f(x) \ = \ \Delta_{h_1}(\Delta_{h_2}( \cdots (\Delta_{h_s} f(x))\cdots)) \ \ \ \mathrm{where} \ \ x, h_1, \ldots, h_s \in \mathbb{K}. \end{align*} $$

When $h = (h_1, \ldots , h_s) \in \mathbb {K}^s$ , we often write $\Delta _{h_1, \ldots , h_s}f(x)$ as $\Delta _h f(x)$ or $\Delta _h^s f(x)$ . For $\omega = (\omega _1, \ldots , \omega _s) \in \{0,1\}^s$ , we write $\omega \cdot h := \sum _{i=1}^s \omega _i h_i$ and $|\omega |:= \omega _1 + \cdots + \omega _s$ . If ${\mathcal C} z = {\overline {z}}$ denotes the conjugation operator, we observe that

(4.2) $$ \begin{align} \Delta_h f(x) \ = \ \prod_{\omega \in \{0,1\}^s} {\mathcal C}^{|\omega|} f(x + \omega \cdot h). \end{align} $$

For any integer $s\ge 1$ , we define the Gowers $U^s$ norm of f by

$$ \begin{align*}\|f\|_{U^s}^{2^s} \ = \ \int_{\mathbb{K}^{s+1} }\Delta_{h_1, \ldots, h_s} f(x) \ d\mu(h_1) \cdots d\mu(h_s) d\mu(x). \end{align*} $$

We note that $\|f\|_{U^2} = \|{\widehat {f}}\|_{L^4}$ .

For intervals $I, I_1, \ldots , I_s$ , we define the Gowers box norm as

$$ \begin{align*} \|f \|_{\square^s_{I_1, \ldots, I_s}(I)}^{2^s} \ = \ \frac{1}{\mu(I)} \int_{\mathbb{K}^{s+1} }\Delta_{h_1, \ldots, h_s} f(x) \ d\nu_{I_1}(h_1) \cdots d\nu_{I_s}(h_s) d\mu(x). \end{align*} $$

From Equation (4.2), we see that

(4.3) $$ \begin{align} \|f\|_{\square_{I_1, \ldots, I_{s+1}}^{s+1}(I)}^{2^{s+1}} \ = \ \int_{\mathbb{K}}\|\Delta_h f\|_{\square_{I_1, \ldots, I_s}^s(I)}^{2^s}d\nu_{I_{s+1}}(h). \end{align} $$

A similar formula relates the Gowers $U^{s+1}$ norm to the Gowers $U^s$ norm.

4.6 The Gowers–Cauchy–Schwarz inequality

When $s\ge 2$ , both the Gowers uniformity norm and the Gowers box norm are in fact norms. In particular, the triangle inequality holds. The triangle inequality also holds when $s=1$ and so we have that

(4.4) $$ \begin{align} \|f + g\|_{U^s} \le \|f\|_{U^s} + \|g\|_{U^s} \ \ \ \mathrm{and} \ \ \ \|f + g\|_{\square_{I_1, \ldots, I_{s}}^{s}(I)} \le \|f\|_{\square_{I_1, \ldots, I_{s}}^{s}(I)} + \|g\|_{\square_{I_1, \ldots, I_{s}}^{s}(I)} \end{align} $$

holds for every $s\ge 1$ . These inequalities follow from a more general inequality which we will find useful.

Let A be a finite set and for each $\alpha \in A$ ; let $(X_{\alpha }, du_{\alpha })$ be a probability space. Set $X = \prod _{\alpha \in A} X_{\alpha }$ , and let $f : X \to {\mathbb C}$ be a complex-valued function. For any $x^{(0)} = (x_{\alpha }^{(0)})_{\alpha \in A}$ and $x^{(1)} = (x_{\alpha }^{(1)})_{\alpha \in A}$ in X and $\omega = (\omega _{\alpha })_{\alpha \in A} \in \{0,1\}^A$ , we write $x^{(\omega )} = (x_{\alpha }^{(\omega _{\alpha })})_{\alpha \in A}$ . We define the generalised Gowers box norm of f on X as

$$ \begin{align*} \|f\|_{\square (X)}^{2^{|A|}} \ = \ \iint_{X^2} \prod_{\omega \in \{0,1\}^A} {\mathcal C}^{|\omega|} f(x^{(\omega)}) \ du(x^{(0)}) \, du(x^{(1)}), \end{align*} $$

where $du$ denotes the product measure $\otimes _{\alpha \in A} du_{\alpha }$ . The following lemma is established in [16].

Lemma 4.5 (Gowers–Cauchy–Schwarz inequality).

With the setup above, let $f_{\omega } : X \to {\mathbb C}$ for every $\omega \in \{0,1\}^A$ . We have

(4.6) $$ \begin{align} \Big| \iint_{X^2} \prod_{\omega \in \{0,1\}^A} {\mathcal C}^{|\omega|} f_{\omega} (x^{(\omega)}) \ du(x^{(0)}) \, du(x^{(1)}) \Big| \ \le \ \prod_{\omega \in \{0,1\}^A} \|f_{\omega}\|_{{\square (X)}}. \end{align} $$

We will need the following consequence.

Corollary 4.7. Let $f : X \to {\mathbb C}$ and for each $\alpha \in A$ , suppose $g_{\alpha }: X \to {\mathbb C}$ is a 1-bounded function that is independent of the $x_{\alpha }$ variable. Then

(4.8) $$ \begin{align} \Big| \int_X f(x) \prod_{\alpha \in A} g_{\alpha}(x) du(x) \Big|^{2^{|A|}} \ \le \ \int_{X^2} \prod_{\omega\in \{0,1\}^A} {\mathcal C}^{|\omega|} f(x^{(\omega)}) \, du(x^{(0)}) du(x^{(1)}). \end{align} $$

Proof. For $\omega ^0 = (0, \ldots , 0)$ , set $f_{\omega ^0} = f$ and for $\omega ^{\beta } = (\omega _{\alpha })_{\alpha \in A}$ with $\omega _{\alpha } = 0$ when $\alpha \not = \beta $ and $\omega _{\beta } = 1$ , set $f_{\omega ^{\beta }} = {\overline {g_{\beta }}}$ . For all other choices of $\omega \in \{0,1\}^A$ , set $f_{\omega } = 1$ . Hence,

$$ \begin{align*}\prod_{\omega\in \{0,1\}^A}{\mathcal C}^{|\omega|} f_{\omega}(x^{(\omega)}) = f(x^{(0)}) \prod_{\alpha \in A} g_{\alpha}(x^{(0)}) \end{align*} $$

since $g_{\alpha }$ is independent of the $\alpha $ variable. Therefore, the inequality (4.6) implies

$$ \begin{align*}\Big| \int_X f(x) \prod_{\alpha \in A} g_{\alpha}(x) du(x) \Big| \le \prod_{\omega\in \{0,1\}^A} \|f_{\omega}\|_{{\square (X)}} \le \|f\|_{{\square (X)}} \end{align*} $$

by the 1-boundedness of each $g_{\alpha }$ . This proves Equation (4.8).

5 Some preliminaries

In this section, we establish a few useful results which we will need in our arguments.

5.1 $U^2$ -inverse theorem

We will use the following inverse theorem for the Gowers box norms.

Lemma 5.1 ( $U^2$ -inverse theorem).

Let $H_1$ and $H_2$ be two scales, and let f be a 1-bounded function supported in an interval I. Then

(5.2) $$ \begin{align} \|f\|_{\square^2_{[H_1],[H_2]}(I)}^4 \ \le \ (H_1 H_2)^{-1} \, \|{\widehat{f}}\|_{L^{\infty}(\mathbb{K})}^2. \end{align} $$

Proof. We have

$$ \begin{align*} \|f\|_{\square^2_{[H_1],[H_2]}(I)}^4 \ = \ \frac{1}{\mu(I)} {\mathop{\iiint}_{\mathbb{K}^3}} \Delta_{h_1, h_2} f(x) d\nu_{[H_1]}(h_1) d\nu_{[H_2]}(h_2) d\mu(x) \end{align*} $$

$$ \begin{align*}= \ {\mathop{\iint}_{\mathbb{K}^2}} g(h_1, h_2) \, d\nu_{[H_1]}(h_1) d\nu_{[H_2]}(h_2) \ = \ {\mathop{\iint}_{\mathbb{K}^2}} {\widehat{g}}(\xi_1, \xi_2) \, {\overline{{\widehat{\nu_{[H_1]}}}(\xi_1) {\widehat{\nu_{[H_2]}}}(\xi_2)}} \, d\mu(\xi_1) d\mu(\xi_2), \end{align*} $$

where

$$ \begin{align*}g(h_1, h_2) \ = \ \frac{1}{\mu(I)} \int_{\mathbb{K}} \Delta_{h_1, h_2} f(x) \, d\mu(x). \end{align*} $$

Hence,

$$ \begin{align*}\|f\|_{\square^2_{[H_1],[H_2]}(I)}^4 \ \le \ \|{\widehat{\nu_{[H_1]}}}\|_{L^1} \|{\widehat{\nu_{[H_2]}}}\|_{L^1} \ \sup_{{\underline{\xi}}\in \mathbb{K}^2} |{\widehat{g}}(\xi_1, \xi_2)| \end{align*} $$

$$ \begin{align*}= \ \frac{H_1^{-1}H_2^{-1}}{\mu(I)} \ \sup_{{\underline{\xi}}\in \mathbb{K}^2} \Big| {\mathop{\iiint}_{\mathbb{K}^3}} f_{00}(x) {\overline{f_{10}(x+h_1)}} {\overline{f_{01}(x+h_2)}} f_{11}(x+h_1+h_2) \, d\mu(x) d\mu(h_1) d\mu(h_2), \Big| \end{align*} $$

where $f_{00}(x) = f(x) \mathrm {e}(-\xi _1 x - \xi _2 x),$

$$ \begin{align*}\ f_{10}(x) \ = \ f(x) \mathrm{e}(-\xi_1 x), \ f_{01}(x) = f(x) \mathrm{e}(-\xi_2 x) \ \ \mathrm{and} \ \ f_{11}(x) \ = \ f(x). \end{align*} $$

The final equality follows since $|\widehat {\nu }_{[H_j]}(\xi )| = |\widehat {\mu }_{[H_j]}(\xi )|^2$ , and so

by Plancherel’s theorem. Furthermore,

$$ \begin{align*}{\widehat{g}}(\xi_1, \xi_2) \ = \ \frac{1}{\mu(I)} {\mathop{\iiint}_{\mathbb{K}^3}} \Delta_{h_1, h_2} f(x) \, \mathrm{e}(\xi_1 h_1 + \xi_2 h_2) \, d\mu(h_1) d\mu(h_2) d\mu(x). \end{align*} $$

Appealing to the Gowers–Cauchy–Schwarz inequality (4.6), we see that

$$ \begin{align*}\|f\|_{\square^2_{[H_1],[H_2]}(I)}^4 \ \le \ (\mu(I) H_1 H_2)^{-1} \|f\|_{U^2}^4 \ = \ (\mu(I) H_1 H_2)^{-1}\|{\widehat{f}}\|_{L^4}^4 \ \le \ (H_1 H_2)^{-1} \|{\widehat{f}}\|_{L^{\infty}}^2 \end{align*} $$

as desired. The last inequality follows from Plancherel’s theorem, the 1-boundedness of f and $\mathrm {supp}(f) \subset I$ which implies

$$ \begin{align*} \|{\widehat{f}}\|_{L^4}^4 \le \|{\widehat{f}}\|_{L^{\infty}}^2 \|{\widehat{f}}\|_{L^2}^2 = \|{\widehat{f}}\|_{L^{\infty}}^2 \|f\|_{L^2}^2 \le \mu(I) \|{\widehat{f}}\|_{L^{\infty}}^2.\\[-34pt] \end{align*} $$

5.2 van der Corput’s inequality

We will need the following useful inequality.

Lemma 5.3 (van der Corput’s inequality).

Let $\mathfrak g \in L^1(\mathbb {K})$ , and let $J = B_{r_J}(x_J)$ be an interval. Then for any scale H, $0<H\le \mu (J)$ , we have

(5.4) $$ \begin{align} \bigg|\int_{\mathbb{K}} \mathfrak g(y)d\mu_J(y)\bigg|^2\leq \frac{C}{\mu(J)}\int_{\mathbb{K}}\int_{J\cap(J-h)}\Delta_h\mathfrak g(y) d\mu(y) d\nu_{[H]}(h). \end{align} $$

We can take $C = 4$ when $\mathbb {K}$ is Archimedean. When $\mathbb {K}$ is non-Archimedean, we can take $C = 1$ and furthermore, for any $h \in [H]$ so that the above inequality can be expressed as

(5.5) $$ \begin{align} \bigg|\int_{\mathbb{K}} \mathfrak g(y)d\mu_J(y)\bigg|^2\leq \iint_{\mathbb{K}^2}\Delta_h\mathfrak g(y) d\mu_{[H]}(h) d\mu_J(y) \end{align} $$

since $d\nu _{[H]} = d\mu _{[H]}$ in this case.

Proof. We define . By a change of variables and Fubini’s theorem, we note

$$\begin{align*}\int_{\mathbb{K}}\mathfrak g(y) d\mu_J(y) = \frac{1}{\mu(J)}\iint_{\mathbb{K}^2} \mathfrak g_J(y+h) d\mu_{[H]}(h) d\mu(y). \end{align*}$$

The function $y\mapsto \int _{\mathbb {K}}\mathfrak {g}_J(y+h) d\mu _{[H]}(h)$ is supported on the set $J - [H]$ which in turn lies in $B_{2(r_J + H)}(x_J)$ (in the non-Archimedean case, $J-[H] = J$ ). Hence, by the Cauchy–Schwarz inequality and a change of variables, we conclude that

$$ \begin{align*} \bigg|\int_{\mathbb{K}}\mathfrak{g}(y) d\mu_J(y)\bigg|^2 &= \frac{1}{\mu(J)^2}\bigg|\iint_{\mathbb{K}^2} \mathfrak g_J(y+h) d\mu_{[H]}(h) d\mu(y)\bigg|^2\\ &\le 2 \frac{\mu(J)+H}{\mu(J)^2}\iiint_{\mathbb{K}^3} \mathfrak g_J(y+h_1)\overline{\mathfrak g_J(y+h_2)} d\mu_{[H]}(h_1) d\mu_{[H]}(h_2) d\mu(y)\\ &= 2 \frac{\mu(J)+H}{\mu(J)^2}\iint_{\mathbb{K}^2}\kappa_{[H]}(h)\mathfrak g_J(y)\overline{\mathfrak g_J(y+h)} d\mu(h)d\mu(y)\\ &\le 4 \mu(J)^{-1} \int_{\mathbb{K}}\int_{J\cap(J-h)}\mathfrak{g}(y)\overline{\mathfrak{g}(y+h)} d\mu(y) d\nu_{[H]}(h), \end{align*} $$

since . This gives the desired conclusion.

5.3 Preparation for the PET induction scheme

We now give a simple application of van der Corput’s inequality which will be repeatedly applied in the PET induction scheme.

Lemma 5.6. Let $c\ge 1$ , and let $I, J\subset \mathbb {K}$ be two intervals with $\mu (I) = N_0$ . Assume that $\mathfrak g_1\in L^\infty (\mathbb {K})$ and $\mathfrak g_2\in L^\infty (\mathbb {K}^2)$ are $1$ -bounded functions such that

(5.7) $$ \begin{align} \|\mathfrak g_1\|_{L^1(\mathbb{K})}\le N_0, \qquad \text{ and } \qquad \sup_{y\in \mathbb{K}}\|\mathfrak g_2(\cdot, y)\|_{L^1(\mathbb{K})}\le c N_0. \end{align} $$

Suppose H is a scale such that $0<H\le \mu (J)$ . When $\mathbb {K}$ is Archimedean, we have

$$ \begin{align*} &\bigg|\frac{1}{N_0}\iint_{\mathbb{K}^2} \mathfrak g_1(x)\mathfrak g_2(x, y)d\mu_J(y)d\mu(x)\bigg|^2\\ \le 4 \bigg|\frac{1}{N_0}\iiint_{\mathbb{K}^3}&\mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu_J(y)d\nu_{[H]}(h)d\mu(x)\bigg| + 8 c \bigg[\frac{\mu([H])}{\mu(J)}\bigg]^{\theta}, \end{align*} $$

where $\theta = 1$ when $\mathbb {K} = {\mathbb R}$ and $\theta = 1/2$ when $\mathbb {K} = {\mathbb C}$ . When $\mathbb {K}$ is non-Archimedean, this improves to

$$ \begin{align*} &\bigg|\frac{1}{N_0}\iint_{\mathbb{K}^2} \mathfrak g_1(x)\mathfrak g_2(x, y)d\mu_J(y)d\mu(x)\bigg|^2\\ \le \frac{1}{N_0} &\iiint_{\mathbb{K}^3}\mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu_J(y)d\mu_{[H]}(h)d\mu(x). \end{align*} $$

Proof. Applying the Cauchy–Schwarz inequality in the x variable, it follows that

$$ \begin{align*}\bigg|\frac{1}{N_0} \iint_{\mathbb{K}^2}\mathfrak g_1(x)\mathfrak g_2(x, y)d\mu_J(y)d\mu(x)\bigg|^2\leq \frac{1}{N_0} \int_{\mathbb{K}} \bigg|\int_{\mathbb{K}} \mathfrak g_2(x, y)d\mu_J(y)\bigg|^2 d\mu(x) \end{align*} $$

since by Equation (5.7) and the $1$ -boundedness of $\mathfrak g_1$ , we have $\|\mathfrak g_1 \|_{L^2(\mathbb {K})}^2 \le N_0$ . By van der Corput’s inequality in Lemma 5.3, we obtain

$$ \begin{align*} \int_{\mathbb{K}} &\bigg|\int_{\mathbb{K}} \mathfrak g_2(x, y)d\mu_J(y)\bigg|^2d\mu(x)\\ \leq 4 \int_{\mathbb{K}}\int_{\mathbb{K}}\kappa_{[H]}(h)&\frac{1}{\mu(J)}\int_{J\cap(J-h)} \mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu(y)d\mu(h)d\mu(x) \end{align*} $$

when $\mathbb {K}$ is Archimedean. In this case, we have $\mu (J\setminus {[J\cap (J-h)]}) \le 2 \mu ([H])$ when $\mathbb {K} = {\mathbb R}$ and $\mu (J\setminus {[J\cap (J-h)]}) \le 2 \sqrt {\mu ([H]) \mu (J)}$ when $\mathbb {K} = {\mathbb C}$ . Hence,

$$ \begin{align*} \frac{4}{N_0} \int_{\mathbb{K}}\kappa_{[H]}(h)\frac{1}{\mu(J)}\int_{J\setminus (J\cap(J-h))}\int_{\mathbb{K}}|\mathfrak g_2(x, y)| d\mu(x)d\mu(y)d\mu(h)\le 8 c \bigg[\frac{\mu([H])}{\mu(J)}\bigg]^{\theta}. \end{align*} $$

In the last line, we used Fubini’s theorem and Equation (5.7) for $\mathfrak g_2$ . This gives the desired bound when $\mathbb {K}$ is Archimedean.

When $\mathbb {K}$ is non-Archimedean, the bound (5.5) in Lemma 5.3 gives

$$ \begin{align*} &\frac{1}{N_0} \int_{\mathbb{K}} \bigg|\int_{\mathbb{K}} \mathfrak g_2(x, y)d\mu_J(y)\bigg|^2d\mu(x)\\ \leq \frac{1}{N_0} &\iiint_{\mathbb{K}^3}\mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu_J(y)d\mu_{[H]}(h)d\mu(x) \end{align*} $$

which is the desired bound in this case.

The next result is an essential building block of the PET induction scheme, which will be employed in Section 6.

Proposition 5.8. Let $N, N_0>0$ be two scales, I an interval such that $\mu (I) = N_0$ , $m\in \mathbb N$ , , and let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of polynomials. Suppose that $\mathfrak f_0, \mathfrak f_1,\ldots , \mathfrak f_m\in L^0(\mathbb {K})$ are $1$ -bounded functions such that $\|\mathfrak f_i\|_{L^1(\mathbb {K})}\le N_0$ for every .

Let $0<\delta \le 1$ , and suppose that

(5.9) $$ \begin{align} \bigg|\frac{1}{N_0} \iint_{\mathbb{K}^2}\mathfrak f_0(x)\prod_{i=1}^m\mathfrak f_{i}(x-P_i(y))d\mu_{[N]}(y)d\mu(x)\bigg|\ge\delta. \end{align} $$

Then there exists an absolute constant $C\gtrsim _{\mathcal P}1$ such that for all $\delta '\le \delta ^4/C$ we have

(5.10) $$ \begin{align} \bigg|\frac{1}{N_0} \iint_{\mathbb{K}^2}\mathfrak f^{\prime}_0(x)\prod_{i=1}^{m'}\mathfrak f^{\prime}_{i}(x-P^{\prime}_i(y))d\mu_{[N]}(y)d\mu(x)\bigg|\gtrsim_C\delta^2, \end{align} $$

where $m'< 2m$ and ${\mathcal P}':=\{P_1^{\prime },\ldots , P_{m'}^{\prime }\}$ is a new collection of polynomials such that

$$\begin{align*}{\mathcal P}'=\{P_1(y)-P_{i_0}(y), P_1(y+h)-P_{i_0}(y),\ldots, P_m(y)-P_{i_0}(y), P_m(y+h)-P_{i_0}(y)\}, \end{align*}$$

for some $\delta '\delta ^2N/C^2\le |h|\le \delta 'N\le \delta ^4N/C$ , where $P_{m'}^{\prime }(y):=P_m(y)-P_{i_0}(y)$ , and $\{\mathfrak f_0^{\prime },\ldots , \mathfrak f_{m'}^{\prime }\}:=\{\mathfrak f_1, \overline {\mathfrak f_1},\ldots , \mathfrak f_m, \overline {\mathfrak f_m}\}$ with $\mathfrak f_{m'}^{\prime }:=\mathfrak f_{m}$ .

Proof. Let and $C\ge 1$ be a large constant to be determined later. We shall apply Lemma 5.6 with $J=[N]$ , the functions $\mathfrak g_1(x)=\mathfrak f_0(x)$ and $\mathfrak g_2(x, y)=\prod _{i\in \mathbb I}\mathfrak f_{i}(x-P_i(y))$ , and the parameter $H=\delta 'N$ . Note that $\|\mathfrak g_1\|_{L^{\infty }(\mathbb {K})}\le 1$ and $\|\mathfrak g_2\|_{L^{\infty }(\mathbb {K}^2)}\le 1$ since $\|\mathfrak f_i\|_{L^{\infty }(\mathbb {K})}\le 1$ for all $i\in \mathbb I$ . Moreover, $\mathfrak g_1$ and $\mathfrak g_2$ satisfy Equation (5.7). If $\delta '\le \delta ^4/C$ and $C\ge 1$ is sufficiently large, using Lemma 5.6, we conclude

$$ \begin{align*} \bigg|\frac{1}{N_0}\iiint_{\mathbb{K}^3}\mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu_{[N]}(y)d\nu_{[H]}(h) d\mu(x)\bigg| \gtrsim \delta^2. \end{align*} $$

By the pigeonhole principle, there exists $|h|\ge \delta ^2 H/C^2$ so that

$$ \begin{align*} \bigg|\frac{1}{N_0} \iint_{\mathbb{K}^2}\mathfrak g_2(x, y)\overline{\mathfrak g_2(x, y+h)}d\mu_{[N]}(y)d\mu(x)\bigg| \gtrsim \delta^2. \end{align*} $$

We make the change of variables $x\mapsto x+P_{i_0}(y)$ to conclude

$$ \begin{align*} \bigg|\frac{1}{N_0}\iint_{\mathbb{K}^2} \prod_{i\in\mathbb I}\mathfrak f_{i}(x-P_i(y)+P_{i_0}(y))\overline{\mathfrak f_{i}(x-P_i(y+h)+P_{i_0}(y))} d\mu_{[N]}(y)d\mu(x)\bigg|\gtrsim\delta^2. \end{align*} $$

This completes the proof.

6 The $L^\infty $ -inverse theorem

The goal of this section is to present the proof of Theorem 1.5, the key $L^\infty $ -inverse theorem for general polynomials with distinct degrees, which we now restate in a more formal, precise way.

Theorem 6.1 (Inverse theorem for $(m+1)$ -linear forms).

Let $N \ge 1$ be a scale, $m\in \mathbb Z_+$ and $0<\delta \le 1$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of polynomials such that $1\le \deg {P_1}<\ldots <\deg {P_m}$ . Set $N_0 = N^{\deg (P_m)}$ , and let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions supported on an interval $I\subset \mathbb {K}$ of measure $N_0$ . Define an $(m+1)$ -linear form corresponding to the pair $({\mathcal P}; N)$ by

(6.2) $$ \begin{align} \Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m):= \frac{1}{N_0} \int_{\mathbb{K}^2}f_0(x)\prod_{i=1}^mf_{i}(x-P_i(y))d\mu_{[N]}(y)d\mu(x). \end{align} $$

Suppose that

(6.3) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then there exists $N_1\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_1)}$ so that

(6.4) $$ \begin{align} N_0^{-1}\big\| \mu_{[N_1]}*f_1\big\|_{L^1(\mathbb{K})} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}. \end{align} $$

If necessary, we will also write $\Lambda _{{\mathcal P}; N}(f_0,\ldots , f_m)=\Lambda _{{\mathcal P}; N, I}(f_0,\ldots , f_m)$ in order to emphasize that the functions $f_0, f_1,\ldots , f_m$ are supported on I.

Remark

When $\mathbb {K} = {\mathbb C}$ is the complex field, the proof of Theorem 6.1 will also hold if the form $\Lambda _{{\mathcal P}; N}$ is defined with the disc $[N] = {\mathbb D}_{\sqrt {N}}$ replaced by the square

$$ \begin{align*}[N]_{sq} \ := \ \{x + i y \in {\mathbb C} : |x|\le \sqrt{N}, |y| \le \sqrt{N}\}. \end{align*} $$

In this case, the conlusion is $N_0^{-1} \| \mu _{[N_1]_{sq}} * f_1\|_{L^1({\mathbb C})} \gtrsim \delta ^{O_{\mathcal P}(1)}$ . This observation will be needed at one point in the proof of Theorem 1.6.

The proof of Theorem 6.1 breaks into two main steps: First, an application of PET induction to show that whenever

$$\begin{align*}|\Lambda_{\mathcal{P};N}(f_0,f_1,\dots,f_m) | \geq \delta \end{align*}$$

is large, then necessarily $f_m$ has a fairly large $U^s$ norm for an appropriately large $s = s_{\mathcal {P}}$ . Second, an inductive ‘degree-lowering’ step to reduce $U^s$ control to $U^2$ control. We accordingly subdivide the argument into two subsections.

6.1 PET induction

Our first goal is to show that whenever the multilinear form $\Lambda _{{\mathcal P};I}$ is large, necessarily $f_m$ has some fairly large (sufficiently high degree) Gowers box norm. We begin with the definition of $(d,j)$ -admissible polynomials. Recall that for a polynomial $P \in \mathbb {K}[\mathrm {y}]$ , the leading coefficient is denoted by $\ell (P)$ .

Definition 6.5 (The class of $(d,j)$ -admissible polynomials).

Let $N\ge 1$ be a scale, $0<\delta \le 1$ , $d\in \mathbb Z_+$ , and parameters $A_0\ge 1$ and $A\ge 0$ be given. Assume that a finite collection of polynomials ${\mathcal P}$ has degree j, and define $\mathcal {P}_j := \{ P\in {\mathcal P} : \deg (P) = j\}$ . We will say that ${\mathcal P}$ is $(d, j)$ -admissible with tolerance $(A_0, A)$ if the following properties are satisfied:

1. 1. For every $P\in {\mathcal P}_j$ , we have

   (6.6) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(P)| \leq A_0\delta^{-A}N^{d-j}. \end{align} $$

2. 2. Whenever $P, Q\in {\mathcal P}_j$ and $\ell (P) \neq \ell (Q)$ , we have

   (6.7) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(P) - \ell(Q)| \leq A_0\delta^{-A} N^{d-j}. \end{align} $$

3. 3. Whenever $P, Q\in {\mathcal P}_j$ and $P\neq Q$ and $\ell (P) = \ell (Q)$ , we have

   (6.8) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j+1} \leq |\ell(P - Q)| \leq A_0\delta^{-A} N^{d-j+1}, \end{align} $$
   and $\deg (P-Q) = j-1$ .

In the special case where the polynomials in $\mathcal {P}$ are linear, we require that $\ell (P) \neq \ell (Q)$ for each $P, Q\in {\mathcal P}$ . The constants $A_0, A$ will be always independent of $\delta $ and N but may depend on ${\mathcal P}$ . In our applications, the exact values of $A_0, A$ will be unimportant, and then we will simply say that the collection $\mathcal {P}$ is $(d, j)$ -admissible.

Remark 6.9. Under the hypotheses of Theorem 6.1, it is not difficult to see that the collection of polynomials ${\mathcal P}=\{P_1,\ldots , P_m\}$ such that $1\le \deg {P_1}<\ldots <\deg {P_m}=d$ is $(d,d)$ -admissible with the tolerance $(\max \{|\ell (P_m)|^{-1}, |\ell (P_m)|\}, 0)$ . Indeed, condition (6.6) can be easily verified and conditions (6.7) and (6.8) are vacuous as ${\mathcal P}_d=\{P_m\}$ .

The main result of this subsection is the following theorem.

Theorem 6.10 (Gowers box norms control $(m+1)$ -linear forms).

Let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of $(d, d)$ -admissible polynomials such that $1\le \deg {P_1}\le \ldots \le \deg {P_m}=d$ . Let $N, N_0\ge 1$ be two scales, I an interval with measure $N_0$ and $0<\delta \le 1$ be given, and let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions such that $\|f_i\|_{L^1(\mathbb {K})}\le N_0$ for all . If Equation (6.3) is satisfied, then there exists $s:=s_{\mathcal P}\in \mathbb Z_+$ such that

(6.11) $$ \begin{align} \|f_m\|_{\square_{[H_1], \ldots, [H_s]}^s(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align} $$

where $H_i\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_m)}$ for .

The proof of Theorem 6.10 requires a subtle downwards induction based on a repetitive application of Proposition 5.8 on the class of $(d, j)$ -admissible polynomials. To make our induction rigorous, we will assign a weight vector to each collection $\mathcal {P}\subset \mathbb {K}[\mathrm {t}]$ of polynomials.

Definition 6.12 (Weight vector).

For any finite $\mathcal {P}\subset \mathbb {K}[\mathrm {t}]$ , define the weight vector

$$\begin{align*}v(\mathcal{P}) := (v_1, v_2,\dots) \in \mathbb{N}^{\mathbb Z_+}, \end{align*}$$

where

$$\begin{align*}v_j :=v_j({\mathcal P}):= \#\{ \ell(P) : P \in \mathcal{P} \text{ and } \deg(P) = j\}, \end{align*}$$

is the number of distinct leading coefficients of $\mathcal {P}$ of degree $j\in \mathbb Z_+$ .

For example, the weight vector for the family ${\mathcal P}=\{x, 5x, x^2, x^2+x, x^4\}$ is $v({\mathcal P})=(2, 1, 0, 1, 0, 0, \ldots )$ . There is a natural ordering on the set of weight vectors.

Definition 6.13 (Well-ordering on the set of weight vectors).

For any two weight vectors $v({\mathcal P})=(v_1({\mathcal P}),v_2({\mathcal P}),\dots )$ and $ v(\mathcal Q)=(v_j(\mathcal Q),v_j(\mathcal Q),\dots )$ corresponding to finite collections $\mathcal {P}, \mathcal Q\subset \mathbb {K}[\mathrm {t}]$ we define an ordering $\prec $ on the set of weight vectors by declaring that

$$\begin{align*}v({\mathcal P})\prec v(\mathcal Q) \end{align*}$$

if there is a degree $j\in \mathbb Z_+$ such that $v_j({\mathcal P})<v_j(\mathcal Q)$ and $v_k({\mathcal P})=v_k(\mathcal Q)$ for all $k>j$ .

It is a standard fact that $\prec $ is a well ordering, we omit the details.

Proof of Theorem 6.10.

We begin by stating the following claim:

Claim 6.14. Let $N, N_0\ge 1$ be two scales, $0<\delta \le 1$ , $d, m\in \mathbb Z_+$ and be given, and let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of $(d, j)$ -admissible polynomials with tolerance $(A_0, A)$ such that $\deg {P_1}\le \ldots \le \deg {P_m} =j$ . Let I be an interval with $\mu (I) = N_0$ , and let $f_0, f_1,\ldots , f_m \in L^0(\mathbb {K})$ be $1$ -bounded functions such that $\|f_i\|_{L^1(\mathbb {K})}\le N_0$ for all . Suppose that

(6.15) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then there exists a collection ${\mathcal P}':=\{P_1^{\prime },\ldots , P_{m'}^{\prime }\}$ of $(d, j-1)$ -admissible polynomials with tolerance $(A_0^{\prime }, A')$ and $m':=\#{\mathcal P}'$ so that $\deg (P_1^{\prime })\le \ldots \le \deg (P_{m'}^{\prime })=j-1$ , and $1$ -bounded functions $f_0^{\prime }, f_1^{\prime },\ldots , f_{m'}^{\prime }\in L^0(\mathbb {K})$ such that $\|f_i^{\prime }\|_{L^1(\mathbb {K})}\le N_0$ for all with $f_{m'}^{\prime }:=f_m$ and satisfying

(6.16) $$ \begin{align} |\Lambda_{{\mathcal P}'; N}(f_0^{\prime},\ldots, f_{m'}^{\prime})|\gtrsim_{\mathcal P}\delta^{O_{\mathcal P}(1)}. \end{align} $$

The proof of Claim 6.14 will use the polynomial exhaustion technique based on an iterative application of the PET induction scheme from Proposition 5.8. The key steps of this method are gathered in Proposition 6.20. Assuming momentarily that Claim 6.14 is true, we can easily close the argument to prove Theorem 6.10. We begin with a collection of $(d, d)$ -admissible polynomials such that $\deg {P_1}\le \ldots \le \deg {P_m}=d$ and apply our claim $d-1$ times until we reach a collection of $(d,1)$ -admissible linear polynomials $\mathcal L$ with distinct leading terms, which satisfies Equation (6.16) with ${\mathcal P}'={\mathcal L}$ . In the special case where all polynomials are linear matters simplify and can be handled using the next result, Proposition 6.17, which in turn implies Equation (6.11) from Theorem 6.10 as desired.

Proposition 6.17. Let $N, N_0 \ge 1$ be two scales, I an interval with $\mu (I) = N_0$ , $0<\delta \le 1$ , $d, m\in \mathbb Z_+$ be given and let $\mathcal L:=\{L_1,\ldots , L_m\}$ be a collection of $(d,1)$ -admissible linear polynomials. Let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions such that $\|f_i\|_{L^1(\mathbb {K})}\le N_0$ for all . Suppose that

(6.18) $$ \begin{align} |\Lambda_{\mathcal L; N}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then we have

(6.19) $$ \begin{align} \|f_m\|_{\square_{[H_1], \ldots, [H_m]}^m(I)} \gtrsim_{\mathcal L} \delta^{2^{m-1}}, \end{align} $$

where $H_i\simeq \delta ^{O_{\mathcal L}(1)}N^{d}$ for .

In fact Proposition 6.17 is a special case of Theorem 6.10 with the collection of linear polynomials $\mathcal L$ in place of ${\mathcal P}$ .

Proof of Proposition 6.17.

Defining we see that each $L'\in {\mathcal L}'$ is linear with vanishing constant term and

$$ \begin{align*} \Lambda_{\mathcal L; N}(f_0,\ldots, f_m)=\Lambda_{{\mathcal L}'; N}(g_0,\ldots, g_m), \end{align*} $$

where $g_i(x)=\mathrm {T}_{{-L_i(0)}}f_i(x)=f_i(x+L_i(0))$ for each . We now apply Lemma 5.6 with functions $\mathfrak g_1(x)=g_0(x)$ and $\mathfrak g_2(x, y)=\prod _{i=1}^mg_i(x-L_i^{\prime }(y))$ and intervals $J=[N]$ , and a parameter $H=\delta ^{M}N/M$ for some large absolute constant $M\ge 1$ , which will be specified later. Using Lemma 5.6 and changing the variables $x\mapsto x-L_1(y)$ we obtain

$$ \begin{align*} \bigg|\frac{1}{N_0} \iiint_{\mathbb{K}^3}\Delta_{\ell(L_1)h} g_1(x)\prod_{i=2}^m\Delta_{\ell(L_i)h}g_{i}(x-(L_i-L_1)(y))d\mu_{[N]}(y)d\mu(x)d\nu_{[H]}(h)\bigg|\gtrsim_M \delta^2. \end{align*} $$

Applying Lemma 5.6 $m-2$ more times and changing the variables $x\mapsto x-L_m(0)$ , we obtain

$$ \begin{align*} \bigg|\frac{1}{N_0} \int_{\mathbb{K}^{m+1}}\Delta_{u_1h_1}\cdots\Delta_{u_{m-1}h_{m-1}}\Delta_{\ell(L_m)h_m}f_{m}(x) d\nu_{[H]}^{\otimes m}(h_1,\ldots, h_m)d\mu(x)\bigg|\gtrsim_M \delta^{2^{m-1}}, \end{align*} $$

where $u_i:=\ell (L_m)-\ell (L_{i})$ for . By another change of variables we obtain Equation (6.19) with

$$ \begin{align*} H_m= |\ell(L_m)|\delta^M N/M, \quad \text{ and } \quad H_i=|\ell(L_m)-\ell(L_{i})|\delta^M N/M \end{align*} $$

for . Using Equation (6.6) with $P=L_m$ , and Equation (6.7) with $P=L_m$ and $Q=L_i$ we obtain that $H_i\simeq \delta ^{O_{\mathcal L}(1)}N^{d}$ for provided that $M\ge 1$ is sufficiently large. This completes the proof of Proposition 6.17.

Proposition 6.20. Let $N, N_0>0$ be two scales, $0<\delta \le 1$ , $d, m\in \mathbb Z_+$ and be given, and let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of $(d, j)$ -admissible polynomials with tolerance $(A_0, A)$ such that $i =\deg {P_1}\le \ldots \le \deg {P_m} =j$ . Let I be an interval with $\mu (I) = N_0$ , and let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions such that $\|f_i\|_{L^1(\mathbb {K})}\le N_0$ for all . Suppose that

(6.21) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then there exists a collection of polynomials ${\mathcal P}':=\{P_1^{\prime },\ldots , P_{m'}^{\prime }\}$ with $m':=\#{\mathcal P}'<2\#{\mathcal P}$ satisfying $P_{m'}^{\prime }:=P_m-P_1$ and $\deg (P_1^{\prime })\le \ldots \le \deg (P_{m'}^{\prime })$ , and $1$ -bounded functions $f_0^{\prime }, f_1^{\prime },\ldots , f_{m'}^{\prime }\in L^0(\mathbb {K})$ such that $\|f_i^{\prime }\|_{L^1(\mathbb {K})}\le N_0$ for all and satisfying

(6.22) $$ \begin{align} |\Lambda_{{\mathcal P}'; N}(f_0^{\prime},\ldots, f_{m'}^{\prime})|\gtrsim_{\mathcal P}\delta^2. \end{align} $$

We also know that $\{f_0^{\prime }, f_1^{\prime },\ldots , f_{m'}^{\prime }\}=\{f_1,\overline {f_1},\ldots , f_m,\overline {f_m}\}$ with $f_{m'}^{\prime } =f_m$ .

Moreover, $v({\mathcal P}')\prec v({\mathcal P})$ , and one of the following three scenarios occurs.

1. (i) The collection ${\mathcal P}$ is of type I; that is, ${\mathcal P}\neq {\mathcal P}_j$ . In this case, ${\mathcal P}'$ is a $(d, j)$ -admissible collection of polynomials with tolerance $(A_0^{\prime }, A')$ and for some $1\le i \le j-1$ ,

   (6.23) $$ \begin{align} \qquad v({\mathcal P}')=(v_1({\mathcal P}'),\ldots, v_{i-1}({\mathcal P}'), v_i({\mathcal P})-1, v_{i+1}({\mathcal P}), \ldots, v_{j}({\mathcal P}),0, 0, \ldots). \end{align} $$

2. (ii) The collection ${\mathcal P}$ is of type II; that is, ${\mathcal P}={\mathcal P}_j$ and $v_j({\mathcal P})>1$ . In this case, ${\mathcal P}'$ is a $(d, j)$ -admissible collection of polynomials with tolerance $(A_0^{\prime }, A')$ and

   (6.24) $$ \begin{align} v({\mathcal P}')=(v_1({\mathcal P}'),\ldots, v_{j-1}({\mathcal P}'), v_j({\mathcal P})-1, 0,0, \ldots). \end{align} $$

3. (iii) The collection ${\mathcal P}$ is of type III; that is, ${\mathcal P}={\mathcal P}_j$ and $v_j({\mathcal P})=1$ . In this case, ${\mathcal P}'$ is a $(d, j-1)$ -admissible collection of polynomials with tolerance $(A_0^{\prime }, A')$ and

   (6.25) $$ \begin{align} v({\mathcal P}')=(0,\ldots,0, v_{j-1}({\mathcal P}'), 0,0, \ldots). \end{align} $$
   Moreover, the leading coefficients of the polynomials in ${\mathcal P}'$ are pairwise distinct.

The tolerance $(A_0^{\prime }, A')$ of the collection ${\mathcal P}'$ only depends on the tolerance $(A_0, A)$ of the collection ${\mathcal P}$ and is independent of $\delta $ and N.

Using Proposition 6.20, we now prove Claim 6.14.

Proof of Claim 6.14.

We may assume, without loss of generality, that the collection ${\mathcal P}$ from Claim 6.14 is of type I or type II. Then we apply Proposition 6.20 until we reach a collection of polynomials of type III with weight vector $v({\mathcal P})=(0,\ldots ,0, v_{j}({\mathcal P}), 0,0, \ldots )$ , where $v_j({\mathcal P}) = 1$ and such that Equation (6.16) holds. We apply Proposition 6.20 once more to reach a collection of $(d, j-1)$ -admissible polynomials satisfying Equation (6.16). This completes the proof of the claim.

Proof of Proposition 6.20.

Appealing to Proposition 5.8 with $i_0=1$ , we may conclude that there exists a collection of polynomials ${\mathcal P}':=\{P_1^{\prime },\ldots , P_{m'}^{\prime }\}$ with $m'=\#{\mathcal P}'<2\#{\mathcal P}$ and $P_{m'}^{\prime }=P_m-P_1$ such that

$$\begin{align*}{\mathcal P}'=\{P_1(y)-P_{1}(y), P_1(y+h)-P_{1}(y),\ldots, P_m(y)-P_{1}(y), P_m(y+h)-P_{1}(y)\}, \end{align*}$$

for some $\delta '\delta ^2N/C^2\le |h| \le \delta 'N\le \delta ^4N/C$ . Proposition 5.8 also ensures that bound (6.22) holds for certain $1$ -bounded functions $f_0^{\prime }, f_1^{\prime },\ldots , f_{m'}^{\prime }\in L^0(\mathbb {K})$ such that $\|f_i^{\prime }\|_{L^1(\mathbb {K})}\le N_0$ for all and satisfying $\{f_0^{\prime }, f_1^{\prime },\ldots , f_{m'}^{\prime }\}=\{f_1,\overline {f_1},\ldots , f_m,\overline {f_m}\}$ with $f_{m'}^{\prime }=f_m$ . Now, it remains to verify conclusions from (i), (ii) and (iii). For this purpose, we will have to adjust $\delta '\le \delta ^4/C$ , which can be made as small as necessary.

Proof of the conclusion from (i)

Suppose that the collection ${\mathcal P}$ is of type I. Then $i=\deg (P_1)<\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_i({\mathcal P}),\ldots , v_j({\mathcal P}), 0, 0, \ldots )$ . To establish Equation (6.23), we consider three cases. Let $P\in {\mathcal P}$ . If $\deg (P)>i$ , then

(6.26) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=\deg(P), \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P), \end{aligned} \end{align} $$

which yields that $v_k({\mathcal P}')=v_k({\mathcal P})$ for all $k>i$ . If $\deg (P)=i$ and $\ell (P)\neq \ell (P_1)$ , then

(6.27) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=i, \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P)- \ell(P_1). \end{aligned} \end{align} $$

If $\deg (P)=i$ and $\ell (P)=\ell (P_1)$ , then

$$ \begin{align*} \deg(P-P_1)<i, \quad\text{ and } \quad \deg(P(\cdot+h)-P_1)<i. \end{align*} $$

The latter two cases show that $v_{k}({\mathcal P}')\ge 0$ for all and $v_i({\mathcal P}')=v_i({\mathcal P})-1$ . Hence, Equation (6.23) holds. We now show that ${\mathcal P}'$ is $(d, j)$ -admissible.

We begin with verifying Equation (6.6) for $P'\in {\mathcal P}_j^{\prime }$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ and $\varepsilon \in \{0, 1\}$ . By Equations (6.26) and (6.6) for $P\in {\mathcal P}_j$ , we obtain

(6.28) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(P')| \leq A_0\delta^{-A}N^{d-j}. \end{align} $$

We now verify Equation (6.7) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ with $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ . We may write

(6.29) $$ \begin{align} Q_1^{\prime}=Q_1(\cdot+\varepsilon_1 h)-P_1, \qquad \text{ and } \qquad Q_2^{\prime}=Q_2(\cdot+\varepsilon_2 h)-P_1 \end{align} $$

for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ . By Equation (6.26), we have $\ell (Q_1^{\prime })=\ell (Q_1)$ and $\ell (Q_2^{\prime })=\ell (Q_2)$ . Then $\ell (Q_1)\neq \ell (Q_2)$ and by Equation (6.7) for $Q_1, Q_2\in {\mathcal P}_j$ , we deduce

(6.30) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(Q_1^{\prime}) - \ell(Q_2^{\prime})| \leq A_0\delta^{-A} N^{d-j}. \end{align} $$

We finally verify Equation (6.8) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ as in Equation (6.29) such that $Q_1^{\prime }\neq Q_2^{\prime }$ and $\ell (Q_1^{\prime })=\ell (Q_2^{\prime })=\ell $ . By Equation (6.26), we see that $\ell (Q_1)=\ell (Q_2)=\ell $ . Since ${\mathcal P}$ is $(d, j)$ -admissible, using Equation (6.6), we also have

(6.31) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell| \leq A_0\delta^{-A}N^{d-j}. \end{align} $$

Recall that $\delta '\delta ^2N/C^2\le |h| \le \delta 'N$ , where $\delta '>0$ is an arbitrarily small number such that $\delta '\le \delta ^4/C$ . Set $\delta ':=\delta ^{M}(CM)^{-1}$ for a large number $M\ge 1$ , which will be chosen later.

First, suppose $Q_1 = Q_2$ . Then $\varepsilon _1 \not = \varepsilon _2$ and $\deg (Q_1^{\prime }-Q_2^{\prime })=j-1$ . Furthermore, $\ell (Q_1^{\prime }-Q_2^{\prime })= j\ell h(\varepsilon _1-\varepsilon _2)$ implying $|\ell (Q_1^{\prime } - Q_2^{\prime })| = |j \ell h|$ , and so by Equation (6.31),

(6.32) $$ \begin{align} |j|(A_0C^3M)^{-1}\delta^{A+M+2} N^{d-j+1}\le |j\ell h| \le |j| A_0(CM)^{-1}\delta^{M-A} N^{d-j+1}, \end{align} $$

and this verifies Equations (6.8) in the case $Q_1 = Q_2$ .

Now, suppose $Q_1 \not = Q_2$ so that $\deg (Q_1 - Q_2) = j-1$ and Equation (6.8) holds for $\ell (Q_1 - Q_2)$ ; that is,

(6.33) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j+1} \leq |\ell(Q_1-Q_2)| \leq A_0\delta^{-A} N^{d-j+1}. \end{align} $$

Taking $M:=\max \{2A, 2|j|A_0^2\}$ in Equation (6.32), we see that $|j\ell h| \le \frac 12 A_0^{-1}\delta ^{A} N^{d-j+1}$ if $C> 1$ is large enough.

In this case, $\ell (Q_1^{\prime } - Q_2^{\prime }) = \ell (Q_1 - Q_2) + j h\ell (\varepsilon _1-\varepsilon _2)$ and so

$$ \begin{align*} |\ell(Q_1-Q_2)|-|j\ell h|\le |\ell(Q_1^{\prime}-Q_2^{\prime})| \le |\ell(Q_1-Q_2)|+|j\ell h|. \end{align*} $$

From Equation (6.33) and $|j\ell h| \le \frac 12 A_0^{-1} \delta ^A N^{d-j+1}$ , we conclude

(6.34) $$ \begin{align} \frac{1}{2}A_0^{-1}\delta^{A} N^{d-j+1} \leq |\ell(Q_1^{\prime}-Q_2^{\prime})| \leq \frac{3}{2}A_0\delta^{-A} N^{d-j+1}. \end{align} $$

This verifies Equation (6.8) in the case $Q_1 \not = Q_2$ .

In either case, we see that $\deg (Q_1^{\prime } - Q_2^{\prime }) = j-1$ and (see Equations (6.32) and (6.34)) we can find a tolerance pair $(A_0^{\prime }, A')$ for ${\mathcal P}'$ depending on the tolerance $(A_0, A)$ of ${\mathcal P}$ and the constants C and M such that

(6.35) $$ \begin{align} (A_0^{\prime})^{-1}\delta^{A'} N^{d-j+1} \leq |\ell(Q_1^{\prime}-Q_2^{\prime})| \leq A_0^{\prime}\delta^{-A'} N^{d-j+1} \end{align} $$

holds, establishing Equation (6.8).

Proof of the conclusion from (ii)

Suppose that the collection ${\mathcal P}$ is of type II. Then $\deg (P_1)=\ldots =\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_j({\mathcal P}), 0, 0, \ldots )$ with $v_j({\mathcal P})>1$ . To establish Equation (6.24), we will proceed in a similar way as in (i). If $P\in {\mathcal P}={\mathcal P}_j$ and $\ell (P)\neq \ell (P_1)$ , then

(6.36) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=j, \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P)- \ell(P_1). \end{aligned} \end{align} $$

If $P\in {\mathcal P}={\mathcal P}_j$ and $\ell (P)=\ell (P_1)$ , then by the fact that ${\mathcal P}$ is $(d, j)$ -admissible, and by Equation (6.8), we see that

(6.37) $$ \begin{align} \deg(P-P_1)<j, \quad\text{ and } \quad \deg(P(\cdot+h)-P_1)<j. \end{align} $$

This shows that $v_{k}({\mathcal P}')\ge 0$ for all and $v_j({\mathcal P}')=v_j({\mathcal P})-1$ . Hence, Equation (6.24) holds. We now show that ${\mathcal P}'$ is $(d, j)$ -admissible.

We begin with verifying Equation (6.6) for $P'\in {\mathcal P}_j^{\prime }$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ such that $\ell (P)\neq \ell (P_1)$ and $\varepsilon \in \{0, 1\}$ . Since ${\mathcal P}$ is $(d, j)$ -admissible, using Equations (6.36) and (6.7) (with $\ell (P)- \ell (P_1)$ in place of $\ell (P)- \ell (Q)$ ), we obtain Equation (6.28) which is Equation (6.6) for $P' \in \mathcal P_j^{\prime }$ .

We now verify Equation (6.7) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ with $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ . As in Equation (6.29), we may write $Q_1^{\prime }=Q_1(\cdot +\varepsilon _1 h)-P_1$ , and $Q_2^{\prime }=Q_2(\cdot +\varepsilon _2 h)-P_1$ for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ such that $\ell (Q_1)\neq \ell (P_1)$ and $\ell (Q_2)\neq \ell (P_1)$ . By Equation (6.36), we have $\ell (Q_1^{\prime })=\ell (Q_1)-\ell (P_1)$ and $\ell (Q_2^{\prime })=\ell (Q_2)-\ell (P_1)$ . Then $\ell (Q_1)\neq \ell (Q_2)$ and Equation (6.30) is verified by appealing to Equation (6.7) (with $\ell (Q_1) - \ell (Q_2)$ in place of $\ell (P) - \ell (Q)$ ).

We finally verify Equation (6.8) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ as in Equation (6.29) such that $Q_1^{\prime }\neq Q_2^{\prime }$ and $\ell (Q_1^{\prime })=\ell (Q_2^{\prime })=\ell $ . By Equation (6.36), $\ell (Q_1)-\ell (P_1)=\ell (Q_2)-\ell (P_1)=\ell $ and since ${\mathcal P}$ is $(d,j)$ -admissible, we see that $\ell $ satisfies Equation (6.31). Now, by following the last part of the proof from (i), we conclude that Equation (6.35) holds.

Proof of the conclusion from (iii)

Suppose that the collection ${\mathcal P}$ is of type III. Then $\deg (P_1)=\ldots =\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_j({\mathcal P}), 0, 0, \ldots )$ with $v_j({\mathcal P})=1$ , thus $\ell (P_1)=\ldots =\ell (P_m):=\ell $ . To establish Equation (6.25), we will proceed in a similar way as in (i) and (ii). If $P\in {\mathcal P}_j$ and $\ell (P)=\ell $ , then Equation (6.31) holds for $\ell $ and once again Equation (6.37) holds. This in turn implies that $v_{j-1}({\mathcal P}')>0$ and $v_{k}({\mathcal P}')=0$ for all $k\neq j-1$ . Hence, Equation (6.25) holds. We now show that ${\mathcal P}'$ is $(d, j-1)$ -admissible.

We begin with verifying Equation (6.6) (or equivalently Equation (6.28) with j replaced by $j-1$ ) for $P'\in {\mathcal P}_{j-1}'$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ such that $\ell (P)=\ell (P_1)$ and $\varepsilon \in \{0, 1\}$ . Then

(6.38) $$ \begin{align} \ell(P')=\ell(P(\cdot+\varepsilon h)-P_1)= \ell(P-P_1) + j h \ell \varepsilon. \end{align} $$

As in (i) we have $\delta '\delta ^2N/C^2\le |h| \le \delta 'N$ , where $\delta ':=\delta ^{M}(CM)^{-1}$ for a large number $M\ge 1$ , which will be chosen later. Furthermore if $P \not = P_1$ , then $A_0^{-1} \delta ^A N^{d-j+1} \le |\ell (P-P_1)| \le A_0 \delta ^{-A} N^{d-j+1}$ since ${\mathcal P}$ is $(d,j)$ -admissible and so Equation (6.8) holds with $Q = P_1$ . This takes care of the case $\varepsilon = 0$ .

If $\varepsilon =1$ and $P = P_1$ , then Equation (6.32) gives the desired bound for $|\ell (P')|$ . When $P \not = P_1$ , we use the upper bound from Equation (6.32)

(6.39) $$ \begin{align} |jh\ell| \leq |j|A_0(CM)^{-1}\delta^{M-A}N^{d-j+1} \le \frac12 A_0^{-1} \delta^{-A} N^{d-j+1} \end{align} $$

when $M = \max (2A, 2|j|A_0^2)$ and $C>1$ chosen large enough. Thus, as before, condition (6.6) holds for $P'$ with some tolerance pair $(A_0^{\prime }, A')$ as desired.

For $Q_1^{\prime } \not = Q_2^{\prime }\in {\mathcal P}^{\prime }_{j-1}$ , we may write $Q_1^{\prime }=Q_1(\cdot +\varepsilon _1 h)-P_1$ , and $Q_2^{\prime }=Q_2(\cdot +\varepsilon _2 h)-P_1$ for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ such that $\ell (Q_1)=\ell (Q_2) = \ell (P_1) = \ell $ . We have $\ell (Q_1 - P_1) - \ell (Q_2 - P_1) = \ell (Q_1 - Q_2)$ and so by Equation (6.38),

(6.40) $$ \begin{align} \ell(Q_1^{\prime})-\ell(Q_2^{\prime})= \ell(Q_1 - Q_2) + j h \ell (\varepsilon_1 - \varepsilon_2). \end{align} $$

We consider two cases.

If $Q_1 = Q_2$ , then necessarily $|\varepsilon _1 - \varepsilon _2| = 1$ and so $\ell (Q_1^{\prime }) \not = \ell (Q_2^{\prime })$ , $\deg (Q_1^{\prime } - Q_2^{\prime }) = j-1$ , and Equation (6.32) shows that Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime } \in {\mathcal P}^{\prime }_{j-1}$ .

If $Q_1 \not = Q_2$ , then $A_0^{-1} \delta ^A N^{d-j+1} \le |\ell (Q_1-Q_2)| \le A_0 \delta ^{-A} N^{d-j+1}$ since ${\mathcal P}$ is $(d,j)$ -admissible, and so Equation (6.8) holds with $P=Q_1 $ and $Q=Q_2$ . From Equation (6.39), we see that $\ell (Q_1^{\prime }) \not = \ell (Q_2^{\prime })$ and Equation (6.40) implies that Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_{j-1}$ .

In either case, we see that Equation (6.8) is vacuously satisfied by ${\mathcal P}'$ and Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime } \in {\mathcal P}^{\prime }_{j-1}$ with (necessarily) $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ .

Concluding, we are able to find a tolerance pair $(A_0^{\prime }, A')$ for ${\mathcal P}'$ depending on the tolerance $(A_0, A)$ of ${\mathcal P}$ and the constants C and M such that the required estimates for Equations (6.38) and (6.40) hold. This completes the proof of Proposition 6.20.

6.2 Degree-lowering

Here, we establish a modulated version of the inverse theorem, which will imply Theorem 6.1.

Theorem 6.41 (Inverse theorem for modulated $(m+1)$ -linear forms).

Let $N\ge 1$ be a scale, and let $0<\delta \le 1$ , $m\in \mathbb Z_+$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that

$$ \begin{align*} 1\le \deg{P_1}<\ldots<\deg{P_{m}}<\deg{Q_1}<\ldots<\deg{Q_n}. \end{align*} $$

Let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . For $n\in \mathbb Z_+$ , we define an $(m+1)$ -linear form corresponding to the triple $({\mathcal P}, \mathcal Q; N)$ and a frequency vector $\xi =(\xi _1,\ldots , \xi _n)\in \mathbb {K}^n$ by

(6.42) $$ \begin{align} \Lambda_{{\mathcal P}; N}^{\mathcal Q; \xi}(f_0,\ldots, f_m):=\frac{1}{N_0} \int_{\mathbb{K}^2}f_0(x)\prod_{i=1}^mf_{i}(x-P_i(y))\mathrm{e}\Big(\sum_{j=1}^n\xi_jQ_j(y)\Big)d\mu_{[N]}(y)d\mu(x). \end{align} $$

For $n=0$ , we set $\mathcal Q=\emptyset $ and we simply write $\Lambda _{{\mathcal P}; N}^{\mathcal Q; \xi }(f_0,\ldots , f_m):=\Lambda _{{\mathcal P}; N}(f_0,\ldots , f_m)$ as in Equation (6.2). Suppose that

(6.43) $$ \begin{align} |\Lambda_{{\mathcal P}; N}^{\mathcal Q; \xi}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then there exists a $C_1 = C_1({\mathcal P}) \gg 1$ such that

(6.44) $$ \begin{align} N_0^{-1}\big\| \mu_{[N_1]}*f_1\big\|_{L^1(\mathbb{K})} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align} $$

for any $N_1 = \delta ^C N^{\deg {P_1}}$ with $C\ge C_1$ .

If necessary we will also write $\Lambda _{{\mathcal P}; N}^{\mathcal Q; \xi }(f_0,\ldots , f_m)=\Lambda _{{\mathcal P}; N, I}^{\mathcal Q; \xi }(f_0,\ldots , f_m)$ in order to emphasise that the functions $f_0, f_1,\ldots , f_m$ are supported on I.

We first show how the Gowers box norms control the dual functions. The dual function, or more precisely the m-th dual function, corresponding to Equation (6.42) is defined as

(6.45) $$ \begin{align} F_m^{\xi}(x):=\int_{\mathbb{K}} F_{m; y}^{\xi}(x) d\mu_{[N]}(y), \qquad x\in \mathbb{K}, \end{align} $$

where

(6.46) $$ \begin{align} F_{m; y}^{\xi}(x):=f_0(x+P_m(y))\prod_{i=1}^{m-1}f_{i}(x-P_i(y)+P_m(y))\mathrm{e}\Big(\sum_{j=1}^n\xi_jQ_j(y)\Big). \end{align} $$

Proposition 6.47 (Gowers box norms control the dual functions).

Let $N\ge 1$ be a scale, and let $0<\delta \le 1$ , $d, m\in \mathbb Z_+$ with $m\ge 2$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that ${\mathcal P}$ is $(d, d)$ -admissible and

$$ \begin{align*}1\le \deg{P_1}\le\ldots\le\deg{P_{m}}\le\deg{Q_1}\le\ldots\le\deg{Q_n}.\end{align*} $$

Let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45). Suppose that Equation (6.43) is satisfied. Then for the exponent $s\in \mathbb Z_+$ which appears in the conclusion of Theorem 6.10, we have

(6.48) $$ \begin{align} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_{s+1}]}^{s+1}(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align} $$

where $H_i\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_m)}$ for .

Proof. By changing the variables $x\mapsto x+P_m(y)$ in Equation (6.42), we may write

$$ \begin{align*} \Lambda_{{\mathcal P}; N}^{\mathcal Q; \xi}(f_0,\ldots, f_m)= \frac{1}{N_0}\int_{\mathbb{K}}\Big(\int_{\mathbb{K}} F_{m; y}^{\xi}(x) d\mu_{[N]}(y)\Big)f_m(x) d\mu(x). \end{align*} $$

By the Cauchy–Schwarz inequality (observing once again that $\|f_m\|_{L^2(\mathbb {K})}^2 \le N_0$ ), we have

$$ \begin{align*} \delta^2 &\le \frac{1}{N_0} \int_{\mathbb{K}} \Big|\int_{\mathbb{K}} F_{m; y}^{\xi}(x)d\mu_{[N]}(y)\Big|^2 d\mu(x) \\ & = \frac{1}{N_0}\bigg|\int_{\mathbb{K}^3}F_{m; y_1}^{\xi}(x)\overline{F_{m; y_2}^{\xi}(x)}d\mu_{[N]}^{\otimes2}(y_1, y_2)d\mu(x)\bigg| \\ &= |\Lambda_{{\mathcal P}; N}^{\mathcal Q; \xi}(f_0, f_1,\ldots, f_{m-1}, \overline{F_m^{\xi}})|, \end{align*} $$

where in the last step we changed variables $x\mapsto x-P_m(y_1)$ . Denote $g_m:=\overline {F_m^{\xi }}$ , and $g_j:=f_j$ for . Our strategy will be to reduce the matter to Theorem 6.10 with the family ${\mathcal P}$ . Observe that $g_j$ is a $1$ -bounded function and $\|g_j\|_{L^1(\mathbb {K})}\lesssim N_0$ for all . Changing the variables $x \mapsto x+h$ in the definition of $\Lambda _{{\mathcal P}; N}^{\mathcal Q; \xi }$ and averaging over $h\in [H_{s+1}]$ where $H_{s+1} =\delta ^{O(1)}N^{\mathrm {deg} P_m}$ , we have

$$ \begin{align*} \delta^4&\le |\Lambda_{{\mathcal P}; N}^{\mathcal Q; \xi}(g_0,\ldots, g_m)|^2\\ &\lesssim \frac{1}{N_0}\int_{\mathbb{K}^2}\Big|\int_{\mathbb{K}}g_0(x+h)\prod_{i=1}^mg_{i}(x+h-P_i(y)) d\mu_{[H_{s+1}]}(h)\Big|^2 d\mu_{[N]}(y)d\mu(x), \end{align*} $$

where in the last line we have used the Cauchy–Schwarz inequality in the x and y variables, noting that $x \to g_0(x+h)$ is supported a fixed dilate of I for every $h\in [H_{s+1}]$ . By another change of variables, we obtain

$$ \begin{align*} \int_{\mathbb{K}}\Lambda_{{\mathcal P}; N}(\Delta_hg_0,\ldots, \Delta_hg_m)d\nu_{[H_{s+1}]}(h)\gtrsim \delta^4. \end{align*} $$

Now, we may find a measurable set $X \subseteq [H_{s+1}]$ such that

$$ \begin{align*} |\Lambda_{{\mathcal P}; N}(\Delta_hg_0,\ldots, \Delta_hg_m)|\gtrsim \delta^4 \end{align*} $$

for all $h\in X$ and $\nu _{[H_{s+1}]}(X)\gtrsim \delta ^4$ . Since $\Delta _hg_j$ is a $1$ -bounded function and $\|\Delta _hg_j\|_{L^1(\mathbb {K})}\lesssim N_0$ for all , we may invoke Theorem 6.10 and conclude that

$$ \begin{align*} \|\Delta_hF_m^{\xi}\|_{\square_{[H_1], \ldots, [H_s]}^s(I)}=\|\Delta_hg_m\|_{\square_{[H_1], \ldots, [H_s]}^s(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)} \end{align*} $$

for all $h\in X$ , where $H_i\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_m)}$ for . Averaging over $h\in X$ and using $\nu _{[H_{s+1}]}(X)\gtrsim \delta ^4$ , we obtain

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_{s+1}]}^{s+1}(I)}^{2^{s+1}} \ = \ \int_{\mathbb{K}}\|\Delta_hF_m^{\xi}\|_{\square_{[H_1], \ldots, [H_s]}^s(I)}^{2^s}d\nu_{[H_{s+1}]}(h) \ \gtrsim_{\mathcal P} \ \delta^{O_{\mathcal P}(1)}, \end{align*} $$

which is Equation (6.48) as desired.

We first establish a simple consequence of the oscillatory integral bound (3.1) which will be important later.

Lemma 6.49. Let $N>1$ be a scale, $m\in \mathbb Z_+$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that

(6.50) $$ \begin{align} 1\le \deg{P_1}<\ldots<\deg{P_{m}}<\deg{Q_1}<\ldots<\deg{Q_n}. \end{align} $$

Define the multiplier corresponding to the families ${\mathcal P}$ and $\mathcal Q$ as follows:

$$ \begin{align*} m_N^{{\mathcal P}, \mathcal Q}(\zeta, \xi):=\int_{\mathbb{K}} e\Big(\sum_{i=1}^m\zeta_iP_i(y)+\sum_{j=1}^n\xi_jQ_j(y)\Big)d\mu_{[N]}(y), \end{align*} $$

where $\zeta =(\zeta _1,\ldots , \zeta _m)\in \mathbb {K}^m$ and $\xi =(\xi _1,\ldots , \xi _n)\in \mathbb {K}^n$ . Let $0<\delta \le 1$ and suppose that

(6.51) $$ \begin{align} |m_N^{{\mathcal P}, \mathcal Q}(\zeta, \xi)|\ge \delta. \end{align} $$

Then there exists a large constant $A\gtrsim _{{\mathcal P}, \mathcal Q} 1$ such that

(6.52)

Proof. Fix an element $\alpha \in \mathbb {K}$ such that $|\alpha | = N$ , and make the change of variables $y \to \alpha y$ to write

$$ \begin{align*}m_N^{{\mathcal P}, \mathcal Q}(\zeta, \xi) \ = \ \int_{B_1(0)} \mathrm{e} \Big(\sum_{i=1}^m\zeta_iP_i(\alpha y)+\sum_{j=1}^n\xi_jQ_j(\alpha y)\Big) d\mu(y). \end{align*} $$

Define $R(y):=\sum _{i=1}^m\zeta _iP_i(y)+\sum _{j=1}^n\xi _jQ_j(y)$ . Then $R(y)$ may be rewritten as

$$ \begin{align*} R(y)=\sum_{l=1}^{\deg{Q_n}}\operatorname{\mathrm{c}}_l(R)y^l \end{align*} $$

The oscillatory integral bound (3.1) implies

(6.53) $$ \begin{align} |m_N^{{\mathcal P}, \mathcal Q}(\zeta, \xi)|\lesssim \bigg(1+\sum_{l=1}^{\deg{Q_n}}|\operatorname{\mathrm{c}}_l(R)|N^l\bigg)^{-1/\deg{Q_n}}. \end{align} $$

Hence, Equation (6.51) implies $\max _l |\operatorname {\mathrm {c}}_l(R)| N^{l} \lesssim \delta ^{-d_{*}}$ , where $d_{*} = \deg {Q_n}$ and the maximum is taken over all . From this, we see that for any sufficiently large $A\ge d_{*}$ ,

(6.54) $$ \begin{align} |\operatorname{\mathrm{c}}_l(R)|N^l\le \delta^{-A}/A \end{align} $$

for all .

Using Equation (6.50), we observe that

(6.55)

(6.56)

Using Equation (6.55) for $j=n$ , we see that Equation (6.54) implies Equation (6.52) for $N^{\deg {Q_n}} |\xi _n|$ . Inductively, we now deduce, using Equation (6.55), that Equation (6.54) implies that Equation (6.52) holds for all $N^{\deg {Q_j}} |\xi _j|$ , . Similarly, using Equations (6.56) and (6.54), we see that that the second displayed equation in Equation (6.52) holds.

The key ingredient in the proof of Theorem 6.41 will be a degree-lowering argument, which reads as follows.

Theorem 6.57 (Degree-lowering argument).

Let $N\ge 1$ be a scale, and let $0<\delta \le 1$ , $m\in \mathbb Z_+$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that

$$ \begin{align*} 1\le \deg{P_1}<\ldots<\deg{P_{m}}<\deg{Q_1}<\ldots<\deg{Q_n}. \end{align*} $$

For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function from Equation (6.45) corresponding to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . Suppose that for some integer $s\in \mathbb Z_+$ one has

(6.58) $$ \begin{align} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_s]}^s(I)} \ge \delta, \end{align} $$

where $H_i\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg (P_m)}$ for . Then

(6.59) $$ \begin{align} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_{s-1}]}^{s-1}(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}. \end{align} $$

Assuming momentarily Theorem 6.57, we prove Theorem 6.41.

Proof of Theorem 6.41.

Our goal is to prove Equation (6.44) when

$$ \begin{align*}\delta \ \le \ |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,\ldots, f_m)|. \end{align*} $$

The proof is by induction on $m\in \mathbb Z_+$ . We divide the proof into two steps. In the first step, we establish the base case for $m=1$ . In the second step, we will use Theorem 6.57 to establish the inductive step.

Step 1.

Assume that $m=1$ so that $N_0 = N^{\deg {P_1}}$ . For $\zeta \in \mathbb {K}$ and $\xi =(\xi _1,\ldots , \xi _n)\in \mathbb {K}^n$ , we define the multiplier

$$ \begin{align*} m_N(\zeta, \xi):=\int_{\mathbb{K}} e\Big(-\zeta P_1(y) + \sum_{j=1}^n \xi_j Q_j(y)\Big)d\mu_{[N]}(y). \end{align*} $$

We now express

$$ \begin{align*} \Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(g_0,g_1) =N_0^{-1}\int_{\mathbb{K}} \widehat{g_0}(-\zeta) \widehat{g_1}(\zeta) m_N(\zeta, \xi)d\mu(\zeta). \end{align*} $$

Using the Cauchy–Schwarz inequality and Plancherel’s theorem, we see

(6.60) $$ \begin{align} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(g_0,g_1)|\leq N_0^{-1}\| g_0 \|_{L^2(\mathbb{K})} \|g_1\|_{L^2(\mathbb{K})} \sup_{\zeta \in \operatorname{\mathrm{supp}}{(\widehat{g_0}\widehat{g_1})}} |m_N(\zeta, \xi)|. \end{align} $$

When $\mathbb {K}$ is non-Archimedean, let so that . When $\mathbb {K}$ is Archimedean, choose a Schwartz function $\varphi : \mathbb {K} \to \mathbb {K}$ such that

For a scale M, we set $\varphi _M(x) = M^{-1} \varphi (M^{-1} x)$ when $\mathbb {K} = {\mathbb R}$ and when $\mathbb {K} = {\mathbb C}$ , we set $\varphi _{M}(z) = M^{-1} \varphi (M^{-1/2} z)$ . When $\mathbb {K}$ is non-Archimedean, we set .

Consider two scales $M_1\simeq \delta ^{C}N^{\deg {P_1}}$ and $N_1\simeq \delta ^{2C}N^{\deg {P_1}}/C$ . Then we obtain

$$ \begin{align*} \delta\le |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1)|\le |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,\varphi_{M_1}*f_1)|+ |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|. \end{align*} $$

Note that

$$ \begin{align*} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,\varphi_{M_1}*f_1)| \le N_0^{-1}\|f_0\|_{L^{\infty}(\mathbb{K})}\|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})} \le N_0^{-1} \|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})}, \end{align*} $$

and

$$ \begin{align*} \|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})}&\le \|\varphi_{M_1}*\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})}+ \|(\varphi_{M_1}-\varphi_{M_1}*\mu_{[N_1]})*f_1\|_{L^1(\mathbb{K})}\\ &\lesssim \|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})} +C^{-1}\delta^C N_0 \end{align*} $$

since $\varphi _{M_1}-\varphi _{M_1}*\mu _{[N_1]} = 0$ when $\mathbb {K}$ is non-Archimedean and when $\mathbb {K}$ is Archimedean, we have the pointwise bound

$$ \begin{align*} |\varphi_{M_1}(x)-\varphi_{M_1}*\mu_{[N_1]}(x)|\lesssim C^{-1}\delta^C \, M_1^{-1}\big(1+M_1^{-1}|x|\big)^{-10}. \end{align*} $$

If $C\ge 1$ is sufficiently large, then we may write

(6.61) $$ \begin{align} \delta\lesssim |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1)|\le N_0^{-1}\|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})}+ |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|. \end{align} $$

By Equation (6.60), we have that

(6.62) $$ \begin{align} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|\lesssim \sup_{\zeta\in \mathbb{K}: |\zeta|\ge M_1^{-1} } |m_N(\zeta, \xi)| \end{align} $$

since $\| f_0 \|_{L^2(\mathbb {K})}\le N_0^{1/2}$ and $\|f_1\|_{L^2(\mathbb {K})}\le N_0^{1/2}$ . We now prove that

(6.63) $$ \begin{align} \sup_{\zeta\in \mathbb{K}: |\zeta|\ge M_1^{-1} } |m_N(\zeta, \xi)|\lesssim \delta^{2}. \end{align} $$

Suppose that inequality (6.63) does not hold, then one has

$$ \begin{align*} |m_N(\zeta, \xi)|\gtrsim \delta^{2} \end{align*} $$

for some $\zeta \in \mathbb {K}$ so that $|\zeta |\ge M_1^{-1}$ . Then Lemma 6.49 implies $N^{\deg {P_1}} |\zeta | \lesssim \delta ^{-A}$ for some large, fixed $A\gtrsim 1$ by Equation (6.52). Since $M_1 = \delta ^C N^{\deg {P_1}}$ , we have $\delta ^{-C} \lesssim \delta ^{-A}$ which is a contradiction if $C \gg A$ . Thus, Equation (6.63) holds.

Hence, by Equations (6.63), (6.62) and (6.61), we see that

$$ \begin{align*}\delta \lesssim N_0^{-1}\|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})} \end{align*} $$

which establishes Theorem 6.41 when $m=1$ .

Step 2.

We now assume that Theorem 6.41 is true for $m-1$ in place of m for some integer $m\ge 2$ . Using Theorem 6.57, we show that this implies Theorem 6.41 for $m\ge 2$ . Note that bound (6.43) implies inequality (6.48) from Proposition 6.47. Now, by Theorem 6.57 applied $s-2$ times we may conclude that

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], [H_2]}^2(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align*} $$

where $H_1, H_2\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg {P_m}}$ . By Lemma 5.1, we can find a $\xi _0\in \mathbb {K}$ such that

(6.64) $$ \begin{align} N_0^{-1}\big|\widehat{F_m^{\xi}}(\xi_0)\big| \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)} \end{align} $$

since $N_0 = N^{\deg {P_m}}$ . By definitions (6.45) and (6.46) and making the change of variables $x\mapsto x-P_m(y)$ , we may write

$$ \begin{align*} N_0^{-1}\widehat{F_m^{\xi}}(\xi_0)&=N_0^{-1}\iint_{\mathbb{K}^2}F_{m;y}^{\xi}(x)e(-\xi_0 x)d\mu_{[N]}(y)d\mu(x)\\&=N_0^{-1}\iint_{\mathbb{K}^2}\mathrm{M}_{\xi_0}f_0(x)\prod_{i=1}^{m-1}f_{i}(x-P_i(y))e\Big(\xi_0 P_m(y)+\sum_{j=1}^n\xi_jQ_j(y)\Big)d\mu_{[N]}(y)d\mu(x)\\&=M^{-1} \Lambda_{{\mathcal P}'; N}^{\mathcal Q', \xi'}(\mathrm{M}_{\xi_0}f_0, f_1,\ldots, f_{m-1}), \end{align*} $$

where $\mathrm {M}_{\xi _0}f_0(x):=\mathrm {e}(-\xi _0 x)f_0(x), \, {\mathcal P}':={\mathcal P}\setminus \{P_m\}, \, \mathcal Q':=\mathcal Q\cup \{P_m\}, \, \xi ':=(\xi _0, \xi _1,\ldots , \xi _n)\in \mathbb {K}^{n+1}$ and $M:=N^{\deg (P_m)-\deg (P_{m-1})}$ . The parameter $N_0^{\prime } := N_0 M^{-1}$ is what appears in the m-linear form $\Lambda _{{\mathcal P}';N}^{\mathcal Q', \xi '}$ . We note that $N_0^{\prime } = N^{\deg {P_{m-1}}}$ .

Thus, Equation (6.64) implies

$$ \begin{align*} M^{-1} |\Lambda_{{\mathcal P}'; N, I}^{\mathcal Q', \xi'}(\mathrm{M}_{\xi_0}f_0, f_1,\ldots, f_{m-1})|\gtrsim \delta^{O(1)}. \end{align*} $$

By translation invariance, we may assume that all functions $f_0, f_1,\ldots , f_{m-1}$ are supported in $[N_0]$ . We can partition into $L \simeq M$ sets, each with measure $\simeq N_0^{\prime }$ contained in an interval $I_k$ lying in an $O(N_0^{\prime })$ neighbourhood of $E_k$ . Furthermore, $E_k$ is an $O(N_1)$ neighbourhood of a set $F_k$ such that $\mu (E_k\setminus F_k) \lesssim N_1$ and . Here, $N_1 \simeq \delta ^{O_{\mathcal P}(1)} N^{\deg (P_1)}$ . In the non-Archimedean setting, this decomposition is straightforward; in this case, we can take $F_k = E_k = I_k$ . If fact if $N_0 = q^{n_0}$ and $N_0^{\prime } = q^{n_0 - \ell }$ so that $M = q^{\ell }$ , then

$$ \begin{align*} [N_0] \ = \ B_{q^{n_0}}(0) \ = \ \bigcup_{x \in {\mathcal F}} B_{q^{n_0 -\ell}}(x) \end{align*} $$

gives our partition of $[N_0]$ , where ${\mathcal F} = \{ x = \sum _{j=0}^{\ell -1} x_j \pi ^{-n_0 + j} : x_j \in o_{\mathbb {K}}/m_{\mathbb {K}} \}$ . Note $\# {\mathcal F} = q^{\ell } = M$ . When $\mathbb {K} = {\mathbb R}$ , one simply decomposes the interval $[N_0] = [-N_0, N_0]$ into M subintervals of equal length and then extend and shrink to obtain intervals $I_k$ and $F_k$ with the desired properties.

When $\mathbb {K} = {\mathbb C}$ , the set $[N_0]$ is a disc and the decomposition is not as straightforward but not difficult to construct by starting with a mesh of squares of side length $\sqrt {N_0^{\prime }}$ which cover $[N_0]$ . It is important that for this case (when $\mathbb {K} = {\mathbb C}$ ) that we allow the sets $E_k$ and $F_k$ to be general sets (not necessarily intervals) with the above properties. The picture should be clear.

Hence, by changing variables $x \to x+P_1(y)$ and then back again,

where and .

By the pigeonhole principle, there exists such that $\# L_0\gtrsim \delta ^{O_{{\mathcal P}'}(1)}M$ , and for every $k\in L_0$ , we have

$$ \begin{align*} |\Lambda_{{\mathcal P}'; N, I_k}^{\mathcal Q', \xi'}(f_0^k, g^k, f_2^k,\ldots, f_{m-1}^k)|\gtrsim \delta^{O_{{\mathcal P}'}(1)}. \end{align*} $$

By the inductive hypothesis, we have

for every $k\in L_0$ and for every $N_1 = \delta ^C N^{\deg {P_1}}$ with $C \ge C_1({\mathcal P}')$ . Note that

(6.65)

and hence for $C\gg 1$ large enough,

Now, we can sum over $k\in L_0$ , using the bound $\# L_0\gtrsim \delta ^{O(1)}M$ and the pairwise disjoint supports of , we obtain

which by (6.65) yields

$$ \begin{align*} N_0^{-1}\big\|\mu_{[N_1]}*f_1\big\|_{L^1(\mathbb{K})} \gtrsim \ \delta^{O_{\mathcal P}(1)}, \end{align*} $$

as desired.

We now state two auxiliary technical lemmas which will be needed in the proof of Theorem 6.57. For $\omega = (\omega _1, \ldots , \omega _n) \in \{0,1\}^n$ and $h = (h_1, \ldots , h_n) \in \mathbb {K}^n$ , we write $\omega \cdot h = \sum _{i=1}^n \omega _i h_i$ and $1-\omega = (1-\omega _1, \ldots , 1-\omega _n)$ .

Lemma 6.66. Let $N\ge 1$ be a scale, and let $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ , $n\in \mathbb {N}$ and scales $H_1,\ldots , H_n$ with each $H_i\le N$ be given. Assume that $\phi :X\to \mathbb R$ is a measurable function defined on a measurable set $X\subseteq H:=\prod _{i=1}^n [H_i]$ . Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials. For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . Suppose that

(6.67)

Then

(6.68) $$ \begin{align} \int_{\square_n(X)}\bigg|N_0^{-1}\int_{\mathbb{K}} F_m(x; h, h')e(-\psi(h, h')x)d\mu(x)\bigg|^2 d\Big(\bigotimes_{i=1}^{n}\nu_{[H_i]}\Big)^{\otimes2}(h, h') \gtrsim \delta^{O(1)}, \end{align} $$

where

$$ \begin{align*} \square_n(X):=\Big\{(h, h')\in H^2: \omega \cdot h + (1-\omega)\cdot h' \in X \text{ for every } \omega\in\{0, 1\}^n\Big\}, \end{align*} $$

and

$$ \begin{align*} F_m(x; h, h'):=\int_{\mathbb{K}}{\Delta}_{h'-h}^nf_0(x+P_m(y))\prod_{i=1}^{m-1} {\Delta}_{h'-h}^nf_{i}(x-P_i(y)+P_m(y))d\mu_{[N]}(y),\\ \psi(h, h'):=\sum_{\omega\in\{0, 1\}^n}(-1)^{|\omega|}\phi\Big(\omega \cdot h + (1-\omega)\cdot h'\Big). \end{align*} $$

Proof. We shall write ${\boldsymbol{\nu }}_n:=\bigotimes _{i=1}^{n}\nu _{[H_i]}$ . Using Equations (4.2), (6.45) and (6.46), we see that the left-hand side of Equation (6.67) can be written as

$$ \begin{align*} \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n+1}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^n}G_0(x, z, h;\boldsymbol{y}) d{\boldsymbol{\nu }}_n(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n+1}}_{[N]}(\boldsymbol{y}), \end{align*} $$

where for $\boldsymbol{y}=(y_{(\omega , 0)}, y_{(\omega , 1)})_{\omega \in \{0, 1\}^n}\in \mathbb {K}^{2^{n+1}}$ , $x, z\in \mathbb {K}$ and $h\in \mathbb {K}^n$ . We have set

Write elements in X as $(h_1, h)$ with $h_1 \in \mathbb {K}$ , and apply the Cauchy–Schwarz inequality in all but the $h_1$ variable (noting that $(x,z) \to G_0(x, z, h;\boldsymbol{y})$ is supported in a product of intervals of measure $\simeq N_0^2$ ) to conclude

(6.69) $$ \begin{align} \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^{n-1}} H_0(x, z, h;\boldsymbol{y}) d{\boldsymbol{\nu }}_{n-1}(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n}}_{[N]}(\boldsymbol{y})\gtrsim \delta^{O(1)}, \end{align} $$

where

$$ \begin{align*} H_0(x, z, h;\boldsymbol{y}):=\bigg|\int_{\mathbb{K}}G_0^1(x, z, (h_1, h);\boldsymbol{y})d\nu_{[H_1]}(h_1)\bigg|^2, \end{align*} $$

and

for $\boldsymbol{y}=(y_{(1, \omega , 0)}, y_{(1, \omega , 1)})_{(j,\omega )\in \{0, 1\}^{n}}\in \mathbb {K}^{2^{n}}$ and $x, z, h_1\in \mathbb {K}$ , $h\in \mathbb {K}^{n-1}$ . Expanding the square and changing variables $x\mapsto x-h_1$ and $z\mapsto z-h_1$ , we may rewrite Equation (6.69) as

(6.70) $$ \begin{align} \nonumber \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^{n+1}} G_1(x, z, h_1, h_1^{\prime}, h;\boldsymbol{y})d\nu_{[H_1]}^{\otimes2}&(h_1, h_1^{\prime}) d{\boldsymbol{\nu }}_{n-1}(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n}}_{[N]}(\boldsymbol{y})\\&\gtrsim \delta^{O(1)}, \end{align} $$

where

Iteratively, for each $i\in \{2,\ldots , n\}$ , we apply the Cauchy–Schwarz inequality in all but the $h_i$ variable to conclude that

$$ \begin{align*} \frac{1}{N_0^2}\int_{\mathbb{K}^2}\int_{\mathbb{K}^2}\int_{\square_n(X)} G_n(x, z, h, h';y, y') d{\boldsymbol{\nu }}_n^{\otimes2}(h, h')d\mu(x)d\mu(z)d\mu^{\otimes 2}_{[N]}(y, y') \gtrsim \delta^{O(1)}, \end{align*} $$

where

$$ \begin{align*} G_n(x, z, h, h';y, y'):={\Delta}_{h'-h}^nF_{m; y}^{\xi}(x)\overline{{\Delta}_{h'-h}^nF_{m; y'}^{\xi}(z)} \mathrm{e}(-\psi((h, h'))(x-z)). \end{align*} $$

We have arrived at Equation (6.68), completing the proof of the lemma

The following lemma is a slight variant of a result found in [38].

Lemma 6.71. Given a scale $N\ge 1$ , $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ , $n\in \mathbb {N}$ and scales $H_1,\ldots , H_{n+1}$ with each $H_i \le N$ . We assume for every that $\varphi _i:\mathbb {K}^n\to \mathbb {K}$ is a measurable function independent of the variable $h_i$ in a vector $h=(h_1, \ldots , h_n)\in \mathbb {K}^n$ . Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials. For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 = N^{\deg {P_m}}$ . Suppose that

(6.72)

Then

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1],\ldots, [H_{n+1}]}^{n+1}(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}. \end{align*} $$

Proof. We shall write as before ${\boldsymbol{\nu }}_n:=\bigotimes _{i=1}^{n}\nu _{[H_i]}$ and also let ${\boldsymbol{\mu}}_n:=\bigotimes _{i=1}^n\mu _{[H_i]}$ . Expanding the Fejér kernel, we may write the left-hand side of Equation (6.72) as

We apply the Cauchy–Schwarz inequality in the $x, z$ and $h'$ variables and Corollary 4.7 to deduce that

$$ \begin{align*} {\mathcal I}^{2^n}&\le \frac{1}{N_0^{2}}\int_{\mathbb{K}^{3n+2}} \prod_{\omega\in\{0, 1\}^n}\mathcal C^{|\omega|}\big({\Delta}_{h^{(\omega)}-h}^nF_m^{\xi}(x) \overline{{\Delta}_{h^{(\omega)}-h}^nF_m^{\xi}(z)}\big)d{\boldsymbol{\mu}}_n^{\otimes3}(h^{(0)}, h^{(1)}, h)d\mu(x)d\mu(z)\\ =& \frac{1}{N_0^2} \int_{\mathbb{K}^{3n}} {\mathcal A}(x,z, h_n^{\prime}, h^{(0)}, h^{(1)}, h'){{\mathcal B}}(x,z,h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n-1}^{\otimes3}(h^{(0)}, h^{(1)}, h')d\mu_{[H_n]}(h_n^{\prime}), \end{align*} $$

where

$$ \begin{align*} {\mathcal A}(x,z,h_n^{\prime}, h^{(0)}, h^{(1)}, h') :=& \int_{\mathbb{K}^2} \prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( F_m^{\xi}(x + h_n^0 - h_n^{\prime})\\ &\times{\overline{F_m^{\xi}(x+h_n^1 - h_n^{\prime}) F_m^{\xi}(z+ h_n^0 - h_n^{\prime})}} F_m^{\xi}(z + h_n^1 - h_n^{\prime}) \Big)\Big] d\mu_{[H_n]}^{\otimes2}(h_n^0, h_n^1), \end{align*} $$

$$ \begin{align*} {\mathcal B}(x,z,h^{(0)}, h^{(1)}, h'):=\prod_{\omega'\in\{0, 1\}^{n-1}} {\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( |F_m^{\xi}(x)|^2 |F_m^{\xi}(z)|^2 \Big)\Big]. \end{align*} $$

Since ${\mathcal A}\ge 0$ , we see that

$$ \begin{align*} {\mathcal I}^{2^n} \le &N_0^{-2} \int_{\mathbb{K}^{3n}} {\mathcal A}(x,z, h_n^{\prime}, h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n-1}^{\otimes3}(h^{(0)}, h^{(1)}, h')d\mu_{[H_n]}(h_n^{\prime})\\ &= \frac{1}{N_0^2} \int_{\mathbb{K}^{3n+1}} \prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( F_m^{\xi}(x + h_n^0)\overline{F_m^{\xi}(x+h_n^1)}\\ &\hspace{3cm} \times \overline{F_m^{\xi}(z+ h_n^0)} F_m^{\xi}(z + h_n^1) \Big)\Big] d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h^{(0)}, h^{(1)}) d{\boldsymbol{\mu}}_{n-1}(h')\\ &= \frac{1}{N_0^2} \int_{\mathbb{K}^{3n+1}} \mathcal C(x,z,h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h^{(0)}, h^{(1)}) d{\boldsymbol{\mu}}_{n-1}(h'), \end{align*} $$

where

$$\begin{align*}\mathcal C(x,z,h^{(0)}, h^{(1)}, h'):=\prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} {\Delta}_{h_n^1 - h_n^0}\Big( F_m^{\xi}(x) {\overline{F_m^{\xi}(z)}}\Big)\Big]. \end{align*}$$

In the penultimate equality, we made the change of variables $x \to x -h_n^0+h_n^{\prime }$ and $z \to z - h_n^0+ h_n^{\prime }$ . Now, proceeding inductively we see that

$$ \begin{align*}{\mathcal I}^{2^n} \le \frac{1}{N_0^2}\int_{\mathbb{K}^{2n+2}} {\Delta}_{h-h'}^n F_m^{\xi}(x) {\overline{{\Delta}^n_{h-h'} F_m^{\xi}(z)}} d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h, h'). \end{align*} $$

Inserting an extra average in the x variable and using the pigeonhole principle to fix z, it follows that

$$ \begin{align*} {\mathcal I}^{2^n}\le \frac{1}{N_0}\int_{\mathbb{K}^{2n+1}} \overline{{\Delta}_{h-h'}^nF_m^{\xi}(z)}\int_{\mathbb{K}}{\Delta}_{h-h'}^nF_m^{\xi}(x+w)d\mu_{[H_{n+1}]}(w)d{\boldsymbol{\mu}}_n^{\otimes2}(h,h')d\mu(x). \end{align*} $$

To conclude, we apply the Cauchy–Schwarz inequality to double the w variable and so

$$ \begin{align*}\delta^{2^{n+1}} \le {\mathcal I}^{2^{n+1}} \le \frac{1}{N_0} \int_{\mathbb{K}^{2n+3}} {\Delta}_{h-h'}^{n+1} F_m^{\xi}(x) d{\boldsymbol{\mu}}_{n+1}^{\otimes2}(h,h')d\mu(x) = \|F_m^{\xi}\|_{\square_{[H_1],\ldots, [H_{n+1}]}^{n+1}(I)}. \end{align*} $$

This completes the proof of the lemma.

Proof of Theorem 6.57.

The proof is by induction on $m\in \mathbb Z_+$ . The proof will consist of several steps. We begin by establishing the following claim.

Claim 6.73. Let $N\ge 1$ be a scale, $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that

$$ \begin{align*}1\le \deg{P_1}<\ldots<\deg{P_{m}}<\deg{Q_1}<\ldots<\deg{Q_n}. \end{align*} $$

For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . Suppose that

(6.74) $$ \begin{align} N^{-1}_0\big|\widehat{F_m^{\xi}}(\zeta)\big| \ge \delta. \end{align} $$

Then for any sufficiently large constant $C\gtrsim _{{\mathcal P}, \mathcal Q}1$ one has

(6.75)

The proof of Claim 6.73 for each integer $m\ge 2$ is itself part of the inductive proof of Theorem 6.57. In the first step, we prove Claim 6.73 for $m=2$ . In the second step, we show that Claim 6.73 for all integers $m\ge 2$ implies Theorem 6.57, this in particular will establish Theorem 6.57 for $m=2$ . In the third step, we finally show that Claim 6.73 for all integers $m\ge 3$ follows from Claim 6.73 and Theorem 6.57 for $m-1$ . Taken together, this shows that Claim 6.73 and Theorem 6.57 hold for each integer $m\ge 2$ , completing the proof of Theorem 6.57.

Step 1.

We now prove Claim 6.73 for $m=2$ . Here, $N_0 = N^{\deg {P_2}}$ . For $\zeta _1, \zeta _2\in \mathbb {K}$ and $\xi \in \mathbb {K}^n$ , we define the multiplier

$$ \begin{align*} m_N(\zeta_1, \zeta_2, \xi):=\int_{B_1(0)} \mathrm{e}\Big(-\zeta_1P_1(\alpha y)+\zeta_2 P_2(\alpha y)+\sum_{j=1}^n\xi_jQ_j(\alpha y)\Big) d\mu(y), \end{align*} $$

where $\alpha \in \mathbb {K}$ satisfies $|\alpha | = N$ . By definitions (6.45) and (6.46) and making the change of variables $x\mapsto x-P_2(y)$ , we may write

$$ \begin{align*} N_0^{-1}\widehat{F_2^{\xi}}(\zeta_2)&=N_0^{-1}\int_{\mathbb{K}}\int_{\mathbb{K}}F_{2;y}^{\xi}(x)\mathrm{e}(-\zeta_2 x)d\mu_{[N]}(y)d\mu(x)\\ &=N_0^{-1}\int_{\mathbb{K}}\widehat{f}_0(\zeta_2-\zeta_1)\widehat{f}_1(\zeta_1)m_N(\zeta_1, \zeta_2, \xi)d\zeta_1. \end{align*} $$

By the Cauchy–Schwarz inequality and Plancherel’s theorem, we obtain

$$ \begin{align*} \delta\le N_0^{-1}|\widehat{F_2^{\xi}}(\zeta_2)|\lesssim N_0^{-1}\|f_0\|_{L^2(\mathbb{K})} \|f_1\|_{L^2(\mathbb{K})}\sup_{\zeta_1\in \mathbb{K}}|m_N(\zeta_1, \zeta_2, \xi)|, \end{align*} $$

which gives for some $\zeta _1\in \mathbb {K}$ that

$$ \begin{align*} \delta\lesssim |m_N(\zeta_1, \zeta_2, \xi)| \end{align*} $$

since $\|f_0\|_{L^2(\mathbb {K})}, \|f_1\|_{L^2(\mathbb {K})}\lesssim N_0^{1/2}$ . Applying Lemma 6.49 with ${\mathcal P}=\{-P_1, P_2\}$ and $\mathcal Q=\{Q_1,\ldots , Q_n\}$ , we deduce that for every sufficiently large $C\gtrsim 1$ one has

This completes the proof of Claim 6.73 for $m=2$ .

Step 2.

In this step, we show that Claim 6.73 for all integers $m\ge 2$ implies Theorem 6.57. In view of Step 1, this will in particular establish Theorem 6.57 for $m=2$ , which is the base case of our double induction. As before, we shall write ${\boldsymbol{\nu }}_{j}:=\bigotimes _{i=1}^{j}\nu _{[H_i]}$ for any $j\in \mathbb Z_+$ . Recall that $N_0 = N^{\deg (P_m)}$ and note

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_s]}^s(I)}^{2^s}= \int_{\mathbb{K}^{s-2}}\|{\Delta}_h^{s-2}F_m^{\xi}\|_{\square_{[H_{s-1}], [H_s]}^2(I)}^4 d{\boldsymbol{\nu }}_{s-2}(h). \end{align*} $$

By Equation (6.58) and the pigeonhole principle, there exists a measurable set $X\subseteq \prod _{i=1}^{s-2}[H_i]$ so that ${\boldsymbol{\nu }}_{s-2}(X)\gtrsim \delta ^{O(1)}$ , and for all $h \in X$ one has

$$ \begin{align*} \|{\Delta}_h^{s-2}F_m^{\xi}\|_{\square_{[H_{s-1}], [H_s]}^2(I)} \gtrsim\delta^{O(1)}. \end{align*} $$

Here, we used that $\operatorname {\mathrm {supp}} F_m^{\xi }$ is a subset of an interval whose measure is at most $O(N_0)$ . By Lemma 5.1, we have

Next, we claim that there is a countable set ${\mathcal F} \subset \mathbb {K}$ , depending on N and $\delta $ such that

(6.76)

for some absolute constant $C_0\in \mathbb Z_+$ and for all $h\in X$ . When $\mathbb {K}$ is non-Archimedean, we take

$$ \begin{align*}{\mathcal F} \ = \ \bigcup_{M\ge 1} \, \Big\{ z = \sum_{j= -M}^{L-1} z_j \pi^j \in \mathbb{K} : z_j \in o_{\mathbb{K}}/m_{\mathbb{K}}\Big\}, \end{align*} $$

where $N_0 = q^L$ . Let $x \in I = B_{N_0}(x_0)$ . For any $\zeta \in \mathbb {K}$ , we have $\zeta \in B_{N_0^{-1}}(\zeta _0)$ for some $\zeta _0 \in {\mathcal F}$ . Note that

$$ \begin{align*}\mathrm{e}(-\zeta x) \ = \ \mathrm{e}(-x \zeta_0) \, \mathrm{e}(-(x-x_0)(\zeta - \zeta_0)) \, \mathrm{e}(-x_0(\zeta - \zeta_0)) \ = \ \mathrm{e}(-x \zeta_0) \, \mathrm{e}(-x_0(\zeta - \zeta_0)) \end{align*} $$

since $|(x-x_0)(\zeta - \zeta _0)| \le N_0 N_0^{-1} = 1$ and $\mathrm {e} = 1$ on $o_{\mathbb {K}}$ . Therefore, since ${\Delta }_h^{s-2}F_m$ is supported in I whenever $h \in X$ . This shows that Equation (6.76) holds for non-Archimedean fields.

When $\mathbb {K}= {\mathbb R}$ , we take ${\mathcal F} := T_0 {\mathbb Z}$ , where

$$ \begin{align*} T_0:=\delta^{C_0}\big(CN_0\big)^{-1} \end{align*} $$

for a sufficiently large constant $C\gtrsim 1$ . When $\mathbb {K} = {\mathbb C}$ , we take ${\mathcal F} := T_1 {\mathbb Z}^2$ , where $T_1 := \delta ^{C_0} (C \sqrt {N_0})^{-1}$ . By the Lipschitz nature of characters on ${\mathbb R}$ or ${\mathbb C}$ , we again see that Equation (6.76) holds in the Archimedean cases. In particular, there exists a measurable function $\phi : X \to {\mathcal F}$ so that

(6.77)

for all $h\in X$ . If necessary, we may additionally assume that the range of $\phi $ is finite.

By Lemma 6.66, it follows that

$$ \begin{align*} \int_{\square_{s-2}(X)}\bigg|N^{-1}_0\int_{\mathbb{K}} F_m(x; h, h')e(-\psi((h, h'))x)d\mu(x)\bigg|^2 d{\boldsymbol{\nu }}_{s-2}^{\otimes2}(h, h') \gtrsim \delta^{O(1)}, \end{align*} $$

where

$$ \begin{align*} F_m(x; h, h')&:=\int_{\mathbb{K}}{\Delta}_{h'-h}^{s-2}f_0(x+P_m(y))\prod_{i=1}^{m-1}{\Delta}_{h'-h}^{s-2}f_{i}(x-P_i(y)+P_m(y))d\mu_{[N]}(y),\\\psi(h, h')&:=\sum_{\omega\in\{0, 1\}^{s-2}}(-1)^{|\omega|}\phi\Big(\omega\cdot h+(1-\omega)\cdot h')\Big). \end{align*} $$

Thus, by the pigeonhole principle, there exists a measurable set $X_0\subseteq \square _{s-2}(X)$ with ${\boldsymbol{\nu }}_{s-2}^{\otimes 2}(X_0)\gtrsim \delta ^{O(1)}$ such that for every $(h, h')\in X_0$ one has

$$ \begin{align*} \bigg|N^{-\deg(P_m)}\int_{\mathbb{K}} F_m(x; h, h')e(-\psi((h, h'))x)d\mu(x)\bigg| \gtrsim \delta^{O(1)}. \end{align*} $$

By Claim 6.73, there is a $c:=c_{m, s}\ge 1$ such that for each $(h, h')\in X_0$ , one has

$$ \begin{align*} |\psi((h, h'))|\lesssim_{m, s} \delta^{-c} N^{-\deg(P_m)}. \end{align*} $$

By the pigeonhole principle, there exists $h'\in \prod _{i=1}^{s-2}[H_i]$ and a measurable set

$$\begin{align*}X_0(h'):=\big\{h\in X: (h, h')\in X_0\ \text{ and } \ |\psi((h, h'))|\lesssim \delta^{-c} N^{-\deg(P_m)}\big\} \end{align*}$$

satisfying ${\boldsymbol{\nu }}_{s-2}(X_0(h'))\gtrsim \delta ^{O(1)}$ . Since $\psi ((h, h'))\in {\mathcal F}$ , we see that

$$ \begin{align*} X_0(h')\subseteq\bigcup_{k\in {\mathbb{K}}}X_{0}^k(h'), \end{align*} $$

where ${\mathbb {K}} = [{O(\delta ^{-O(1)})}]\cap {\mathbb Z}$ when $\mathbb {K} = {\mathbb R}$ . In this case, $X_{0}^k(h'):=\{h\in X: \psi ((h, h'))=T_0k\}$ . When $\mathbb {K} = {\mathbb C}$ , we have ${\mathbb {K}} = [{O(\delta ^{-O(1)})}]\cap {\mathbb Z}^2$ and $X_{0}^k(h'):=\{h\in X: \psi ((h, h'))=T_1 k\}$ . Finally when $\mathbb {K}$ is non-Archimedean,

$$ \begin{align*}{\mathbb{K}} = \bigl[{O(\delta^{-O(1)})}\bigr] \cap \bigl\{ k = \sum_{j= -M}^{-1} k_j \pi^j \in \mathbb{K} : k_j \in o_{\mathbb{K}}/m_{\mathbb{K}}\bigr\} \end{align*} $$

and $X_{0}^k(h') := \{h\in X: \psi ((h,h')) = \pi ^L k \}$ .

Thus, by the pigeonhole principle there is $k_0\in {\mathbb {K}}$ such that ${\boldsymbol{\nu }}_{s-2}(X_0^{k_0}(h'))\big )\gtrsim \delta ^{O(1)}$ . When $\mathbb {K} = {\mathbb R}$ , this shows that $\psi (h, h')=T_0 k_0 =: \phi _m$ for all $h\in X_0^{k_0}(h')$ . When $\mathbb {K} = {\mathbb C}$ , we have $\phi (h,h') = T_1 k_0$ for all $h \in X_0^{k_0}(h')$ and when $\mathbb {K}$ is non-Archimedean, $\phi (h,h') = \pi ^L k_0$ for all $h \in X_0^{k_0}(h')$ . We will denote these values by $\phi _m$ in all cases.

Set

$$ \begin{align*} \psi_1(h, h'):=(-1)^{s+1}\sum_{\substack{\omega\in\{0,1\}^{s-2} \\ \omega_1=0}}(-1)^{|\omega|}\phi\Big((\omega\cdot h+(1-\omega)\cdot h')\Big) + (-1)^s \phi_m \end{align*} $$

and, for , set

$$ \begin{align*} \psi_i(h, h'):=(-1)^{s+1}\sum_{\substack{\omega\in\{0,1\}^{s-2}\setminus\{0\} \\ \omega_1=\ldots=\omega_{i-1}=1 \\ \omega_{i}=0}}(-1)^{|\omega|}\phi\Big((\omega\cdot h+(1-\omega)\cdot h')\Big). \end{align*} $$

Note that $\psi _i$ does not depend on $h_{i}$ and we can write

$$ \begin{align*} \phi(h)=\sum_{i=1}^{s-2}\psi_i(h, h'). \end{align*} $$

Averaging Equation (6.77) over $\mathbb {X}:=X_0^{k_0}(h')$ and using positivity, we obtain

Invoking Lemma 6.71, we conclude that

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_{s-1}]}^{s-1}(I)}\gtrsim \delta^{O(1)}. \end{align*} $$

Step 3.

Gathering together the conclusions of Step 1 and Step 2 (for $m=2$ ), we see that the base step of a double induction has been established. In this step, we shall illustrate how to establish the inductive step. We assume that Claim 6.73 and Theorem 6.57 hold for $m-1$ in place m for some integer $m\ge 3$ . Then we will prove that Claim 6.73 holds for $m\ge 3$ , which in view of Step 2 will allow us to deduce that Theorem 6.57 also holds for $m\ge 3$ . This will complete the proof of Theorem 6.57.

Recall that $N_0 = N^{\deg (P_m)}$ . By definitions (6.45) and (6.46) and making the change of variables $x\mapsto x-P_m(y)$ , we may write

$$ \begin{align*} N_0^{-1}\widehat{F_m^{\xi}}(\zeta_m)&=N_0^{-1}\int_{\mathbb{K}^2}F_{m;y}^{\xi}(x)\mathrm{e}(-\zeta_m x)d\mu(x)d\mu_{[N]}(y)\\&=N_0^{-1}\int_{\mathbb{K}^2}\mathrm{M}_{\zeta_m}f_0(x)\prod_{i=1}^{m-1}f_{i}(x-P_i(y))\mathrm{e}\Big(\zeta_m P_m(y)+\sum_{j=1}^n\xi_jQ_j(y)\Big)d\mu_{[N]}(y)d\mu(x)\\&=: M^{-1} \Lambda_{{\mathcal P}'; N}^{\mathcal Q', \xi'}(\mathrm{M}_{\zeta_m}f_0, f_1,\ldots f_{m-1}), \end{align*} $$

where $\mathrm {M}_{\zeta _m}f_0(x):=\mathrm {e}(-\zeta _m x)f_0(z)$ , ${\mathcal P}':={\mathcal P}\setminus \{P_m\}$ , $\mathcal Q':=\mathcal Q\cup \{P_m\}$ , $\xi ':=(\zeta _m, \xi _1,\ldots , \xi _n)\in \mathbb {K}^{n+1}$ and $M = N_0 {N_0^{\prime }}^{-1}$ , where $N_0^{\prime }$ is the scale $N^{\deg (P_{m-1})}$ .

Thus, Equation (6.74) implies

$$ \begin{align*} M^{-1}|\Lambda_{{\mathcal P}'; N}^{\mathcal Q', \xi'}(\mathrm{M}_{\zeta_m}f_0, f_1,\ldots, f_{m-1})|\gtrsim \delta^{O(1)}. \end{align*} $$

As in the proof of Theorem 6.1, by the pigeonhole principle, we can find an interval $I'\subset \mathbb {K}$ of measure about $ N_0^{\prime }$ such that

$$ \begin{align*} |\Lambda_{{\mathcal P}'; N, I'}^{\mathcal Q', \xi'}(f_0^{\prime}, f_1^{\prime},\ldots, f_{m-1}')|\gtrsim \delta^{O(1)}, \end{align*} $$

where .

Consequently, by Proposition 6.47, there exists an $s\in \mathbb Z_+$ such that

$$ \begin{align*} \|F_{m-1}^{\xi'}\|_{\square_{[H_1], \ldots, [H_s]}^s(N_0^{\prime})} \gtrsim \delta^{O(1)}, \end{align*} $$

where $F_{m-1}^{\xi '}$ is the dual function respect the form $\Lambda _{{\mathcal P}'; N, I'}^{\mathcal Q', \xi '}(f_0^{\prime }, f_1^{\prime },\ldots , f_{m-1}')$ and $H_i\simeq \delta ^{O_{{\mathcal P}'}(1)}N^{\deg (P_{m-1})}$ for . By the induction hypothesis (for Theorem 6.57), we deduce that

$$ \begin{align*} \|F_{m-1}^{\xi'}\|_{\square_{[H_1], [H_2]}^2(N_0^{\prime})} \gtrsim \delta^{O(1)}, \end{align*} $$

which in turn by Lemma 5.1 implies

$$ \begin{align*} (N^{\prime}_0)^{-1}\big|\widehat{F_{m-1}^{\xi'}}(\zeta_{m-1})\big| \gtrsim \delta^{O(1)} \end{align*} $$

for some $\zeta _{m-1}\in \mathbb {K}$ . By the induction hypothesis (for Claim 6.73), we deduce that

which in particular implies Equation (6.75), and we are done.

7 Sobolev estimates

As a consequence of the $L^\infty $ -inverse theorem from the previous section, we establish some Sobolev estimates, which will be critical in the proof of Theorem 1.3.

We begin with a smooth variant of Theorem 6.1. When $\mathbb {K}$ is Archimedean, we fix a Schwartz function $\varphi $ on $\mathbb {K}$ so that

When $\mathbb {K}={\mathbb R}$ , we set $\varphi _N(x) = N^{-1} \varphi (N^{-1} x)$ for any $N>0$ and when $\mathbb {K} = {\mathbb C}$ , we set $\varphi _{N}(z) = N^{-1} \varphi (N^{-1/2} z)$ for any $N> 0$ . When $\mathbb {K}$ is non-Archimedean, we set so that and we set for any scale N.

Theorem 7.1 (A smooth variant of the inverse theorem).

Let $N\ge 1$ be a scale, $0<\delta \le 1$ , $m\in \mathbb Z_+$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of polynomials such that $1\le \deg {P_1}<\ldots <\deg {P_m}$ . Let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions supported on an interval $I\subset \mathbb {K}$ of measure $N_0 = N^{\deg {P_m}}$ . Suppose that the $(m+1)$ -linear form defined in Equation (6.2) satisfies

(7.2) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\ldots, f_m)|\ge\delta. \end{align} $$

Then for any there exists an absolute constant $C_j\gtrsim _{\mathcal P}1$ so that

(7.3) $$ \begin{align} N_0^{-1}\big\| \varphi_{N_j}*f_j\big\|_{L^1(\mathbb{K})} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align} $$

where $N_j\simeq \delta ^{C_j}N^{\deg (P_j)}$ , provided $N \gtrsim \delta ^{-O_{\mathcal P}(1)}$ .

Proof. By translation invariance, we can assume that $f_j$ is supported on $[N_0]$ for every . The proof will consist of two steps. In the first step, we will invoke Theorem 6.1 to prove Equation (7.3) for $j=1$ . In the second step, we will use Equation (7.3) for $j=1$ to establish Equation (7.3) for $j=2$ , and continuing inductively we will obtain Equation (7.3) for all .

Step 1.

We first establish Equation (7.3) for $j=1$ . When $\mathbb {K}$ is non-Archimedean, this is an immediate consequence of Theorem 6.1 since $\varphi _{N_1} = \mu _{[N_1]}$ in this case. Nevertheless, we make the observation that

(7.4) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\varphi_{N_1}*f_1, \ldots, f_m)|\gtrsim\delta \end{align} $$

holds. In fact, we will see that Equation (7.4) holds for any $\mathbb {K}$ , non-Archimedean or Archimedean. First, let us see Equation (7.4) when $\mathbb {K}$ is non-Archimedean. Suppose that $|\Lambda _{{\mathcal P}; N}(f_0,\varphi _{N_1}*f_1, \ldots , f_m)| \le c \, \delta $ for some small $c>0$ . Then, since

$$ \begin{align*}\delta \le |\Lambda_{{\mathcal P}; N}(f_0,f_1, \ldots, f_m)| \le |\Lambda_{{\mathcal P}; N}(f_0,\varphi_{N_1}*f_1, \ldots, f_m)| + |\Lambda_{{\mathcal P}; N}(f_0,f_1 -\varphi_{N_1}*f_1, \ldots, f_m)|, \end{align*} $$

we conclude that $|\Lambda _{{\mathcal P}; N}(f_0,f_1 - \varphi _{N_1}*f_1, \ldots , f_m)|\gtrsim \delta $ . Therefore, Theorem 6.1 implies that $N_0^{-1} \|\varphi _{N_1} *(f_1 - \varphi _{N_1} * f_1)\|_{L^1(\mathbb {K})}| \gtrsim \delta ^{O(1)}$ , but this is a contradiction since $\varphi _{N_1} * \varphi _{N_1} = \varphi _{N_1}$ when $\mathbb {K}$ is non-Archimedean (in which case ) and so $\varphi _{N_1} * (f_1 - \varphi _{N_1}* f_1) \equiv 0$ .

We now turn to establish (7.3) for $j=1$ when $\mathbb {K}$ is Archimedean (when $\mathbb {K} = {\mathbb R}$ or $\mathbb {K} = {\mathbb C}$ ). Let $\eta :\mathbb {K} \to [0, \infty )$ be a Schwartz function so that $\int _{\mathbb {K}}\eta =1$ , ${\widehat {\eta }}\equiv 1$ near 0 and $\mathrm {supp}\:\widehat {\eta }\subseteq [2]$ . For $t>0$ , we write $\eta _{t}(x):=t^{-1}\eta (t^{-1}x)$ when $\mathbb {K}={\mathbb R}$ and $\eta _t(x):= t^{-2}\eta (t^{-1}x)$ when $\mathbb {K} = {\mathbb C}$ . We will also need a Schwartz function $\rho :\mathbb {K}\to [0, \infty )$ such that

for some large absolute constant $M\ge 1$ , which will be specified later. We shall also write $\rho _{(t)}(x):=\rho (t^{-1}x)$ for $t>0$ and $x\in \mathbb {K}$ .

Let $N_0^{\prime }\simeq N_0$ when $\mathbb {K} = {\mathbb R}$ and $N_0^{\prime } \simeq \sqrt {N_0}$ when $\mathbb {K} = {\mathbb C}$ . Observe that Equation (7.2) implies that at least one of the following lower bounds holds:

(7.5) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\varphi_{N_1}*f_1, \ldots, f_m)|\gtrsim\delta, \end{align} $$

(7.6) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,\rho_{(N_0^{\prime})}(f_1-\varphi_{N_1}*f_1), \ldots, f_m)|\gtrsim\delta, \end{align} $$

(7.7) $$ \begin{align} |\Lambda_{{\mathcal P}; N}(f_0,(1-\rho_{(N_0^{\prime})})(f_1-\varphi_{N_1}*f_1), \ldots, f_m)|\gtrsim\delta. \end{align} $$

By Theorem 6.1, it is easy to see that Equation (7.5) yields that

$$ \begin{align*} N_0^{-1}\big\| \varphi_{N_1}*f_1\big\|_{L^1(\mathbb R)} \gtrsim \delta, \end{align*} $$

which in turn will imply Equation (7.3) for $j=1$ provided that the remaining two alternatives (7.6) and (7.7) do not hold. If this is the case, then Equation (7.4) also holds when $\mathbb {K}={\mathbb R}, {\mathbb C}$ is Archimedean.

If the second alternative holds we, let $f_1^{\prime }:= \rho _{(N_0^{\prime })}(f_1-\varphi _{N_1}*f_1)$ and then Theorem 6.1 implies that

$$ \begin{align*} N_0^{-1}\big\| \mu_{[N_1^{\prime}]}*f_1^{\prime}\big\|_{L^1(\mathbb{K})} \gtrsim_{\mathcal P} \delta^{C_0^{\prime}}, \end{align*} $$

with $N_1^{\prime }\simeq \delta ^{C_1^{\prime }}N^{\deg (P_1)}$ . By the Cauchy–Schwarz inequality (the support of $\mu _{[N_1^{\prime }]}, *f_1^{\prime }$ is contained in a fixed dilate of $[N_0]$ ), we have

$$ \begin{align*} N_0^{-1}\big\| \mu_{[N_1^{\prime}]}*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2 \gtrsim_{\mathcal P} \delta^{2C_0^{\prime}}. \end{align*} $$

Let $N_1^{\prime \prime }:= \delta ^{A+C_1^{\prime }}N^{\deg (P_1)}/A$ for some $A\ge 1$ to be determined later. We now show that

(7.8) $$ \begin{align} \big\| \mu_{[N_1^{\prime}]}-\mu_{[N_1^{\prime}]}*\eta_{N_1^{\prime\prime}}\big\|_{L^1(\mathbb{K})}^2 \lesssim \sqrt{N_1^{\prime\prime} /N_1^{\prime}} \lesssim \sqrt{\delta^{A}/A}. \end{align} $$

We note that for $|x| \ge C N_1^{\prime }$ ,

and so

When $|x| \le C N_1^{\prime }$ is small, we use the Cauchy–Schwarz inequality

and then Plancherel’s theorem,

Here, we use the facts that $\widehat {\eta } \equiv 1$ near $0$ and the Fourier decay bound for Euclidean balls,

This establishes Equation (7.8) and so

$$ \begin{align*} N_0^{-1}\big\| (\mu_{[N_1^{\prime}]}-\mu_{[N_1^{\prime}]}*\eta_{N_1^{\prime\prime}})*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2&\lesssim \big\| \mu_{[N_1^{\prime}]}-\mu_{[N_1^{\prime}]}*\eta_{N_1^{\prime\prime}}\big\|_{L^1(\mathbb{K})}^2\\ &\lesssim \sqrt{N_1^{\prime\prime} /N_1^{\prime}} \lesssim \sqrt{\delta^{A}/A}. \end{align*} $$

Consequently,

$$ \begin{align*} \delta^{2C_0^{\prime}}\lesssim_{\mathcal P}N_0^{-1}\big\| \mu_{[N_1^{\prime}]}*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2\lesssim N_0^{-1}\big\| \mu_{[N_1^{\prime}]}*\eta_{N_1^{\prime\prime}}*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2+ \sqrt{\delta^{A}/A}, \end{align*} $$

which for sufficiently large $A\ge C_0^{\prime }$ yields

$$ \begin{align*} N_0^{-1}\big\| \eta_{N_1^{\prime\prime}}*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2\gtrsim_{\mathcal P} \delta^{2C_0^{\prime}}. \end{align*} $$

Taking $N_1:=\frac {1}{2}N_1^{\prime \prime }$ and using support properties of $\widehat {\varphi }$ and $\widehat {\eta }$ , by the Plancherel theorem we may write (when $\mathbb {K} = {\mathbb R}$ )

$$ \begin{align*} N_0^{-1}\big\| \eta_{N_1^{\prime\prime}}*f_1^{\prime}\big\|_{L^2({\mathbb R})}^2 =N_0^{-1}\big\| \widehat{\eta}_{N_1^{\prime\prime}}\big(\widehat{\rho_{(N_0^{\prime})}}*((1-\widehat{\varphi}_{N_1})\widehat{f}_1)\big)\big\|_{L^2(\mathbb R)}^2\\ \lesssim N_0^{-1}\int_{\mathbb R}\bigg(\int_{\mathbb R}\frac{N_0^{\prime}}{(1+N_0^{\prime}|\xi-\zeta|)^{200}}|\widehat{f}_1(\zeta)(1-\widehat{\varphi}(N_1\zeta))||\widehat{\eta}(N_1^{\prime\prime}\xi)|\bigg)^2d\mu(\xi)\\ &\kern-200pt\lesssim N_0^{-1}\delta^{100(A+C_1^{\prime})}\|f_1\|_{L^2(\mathbb R)}^2. \end{align*} $$

A similar bound holds when $\mathbb {K} = {\mathbb C}$ . Therefore,

$$ \begin{align*} \delta^{2C_0^{\prime}}\lesssim_{\mathcal P} N_0^{-1}\big\| \eta_{N_1^{\prime\prime}}*f_1^{\prime}\big\|_{L^2(\mathbb{K})}^2\lesssim \delta^{100(A+C_1^{\prime})}, \end{align*} $$

which is impossible if $A\ge 1$ is large enough. Thus, the second alternative (7.6) is impossible. To see that the third alternative (7.7) is also impossible observe that

$$ \begin{align*} \delta\lesssim |\Lambda_{{\mathcal P}; N}(f_0,(1-\rho_{(N_0^{\prime})})(f_1-\varphi_{N_1}*f_1), \ldots, f_m)|\lesssim N_0^{-1}\int_{[N_0^{\prime}]}(1-\rho_{(N_0^{\prime})})(x)d\mu(x)\lesssim \delta^M, \end{align*} $$

which is also impossible if $M\ge 1$ is sufficiently large. Hence, Equation (7.5) must necessarily hold, and we are done.

Step 2.

Let $M\ge 1$ be a large constant to be determined later, and define $N'\simeq \delta ^MN$ and $N_0^{\prime } \simeq \delta ^M N_0$ . The main idea is to partition the intervals $[N]$ and $[N_0]$ into $\mathbb {K}\simeq \delta ^{-M}$ disjoint intervals of measure $\simeq N'$ and $\simeq N_0^{\prime }$ , respectively. Such partitions are straightforward when $\mathbb {K} = {\mathbb R}$ . When $\mathbb {K}$ is non-Archimedean, we only need to partition $[N]$ and not $[N_0]$ . Finally, when $\mathbb {K} = {\mathbb C}$ , intervals are discs and it is not possible to partition a disc into subdiscs and so we will need to be careful with this technical issue.

We first concentrate on the case when $\mathbb {K}$ is non-Archimedean. In this case, we only need to partition $[N]$ and not $[N_0]$ . Such a partition was given in the proof of Theorem 6.41. In fact, choosing $\ell \gg 1$ such that $q^{-\ell } \simeq \delta ^M$ and setting $N = q^n$ so that $N' = q^{n-\ell }$ , we have

$$ \begin{align*} [N] \ = \ B_{q^{n}}(0) \ = \ \bigcup_{y \in {\mathcal F}} B_{q^{n -\ell}}(y), \end{align*} $$

which gives a partition of $[N]$ where ${\mathcal F} = \{ y = \sum _{j=0}^{\ell -1} y_j \pi ^{-n + j} : y_j \in o_{\mathbb {K}}/m_{\mathbb {K}} \}$ . Note $\# {\mathcal F} = q^{\ell }$ so that $\# {\mathcal F} \simeq \delta ^{-M}$ . Hence, $\Lambda _{{\mathcal P}; N}(f_0,\varphi _{N_1}*f_1, \ldots , f_m) =$

$$ \begin{align*}\frac{1}{N_0 N} \sum_{y_0 \in {\mathcal F}} \int_{\mathbb{K}} \int_{B_{q^{n-\ell}}(y_0)} f_0(x) \varphi_{N_1} * f_1 (x - P_1(y)) \prod_{i=2}^m f_i(x - P_i(y)) d\mu(y) d\mu(x). \end{align*} $$

We observe that $\varphi _{N_1} * f_1 (x - P_1(y)) = \varphi _{N_1}* f_1(x - P_1(y_0))$ for any $y \in B_{q^{n-\ell }}(y_0)$ by the non-Archimedean nature of $\mathbb {K}$ , if M is chosen large enough depending on $P_1$ . Hence, by the pigeonhole principle, we can find a $y_0 \in {\mathcal F}$ such that

$$ \begin{align*}\Big| \frac{1}{N_0 N'} \int_{\mathbb{K}} \int_{B_{q^{n-\ell}}(y_0)} f_0(x) \varphi_{N_1} * f_1 (x - P_1(y_0)) \prod_{i=2}^m f_i(x - P_i(y)) d\mu(y) d\mu(x) \Big| \gtrsim \delta. \end{align*} $$

Changing variables $y \to y_0 + y$ allows us to write the above as

$$ \begin{align*}|\Lambda_{{\mathcal P}', N'}(f_0^{\prime}, f_2^{\prime},\ldots, f_m^{\prime})| \ \gtrsim \ \delta \ \ \ \mathrm{where} \end{align*} $$

$$ \begin{align*}\Lambda_{{\mathcal P}', N'}(f_0^{\prime}, f_2^{\prime},\ldots, f_m^{\prime}) \ = \frac{1}{N_0} \int_{\mathbb{K}^2} f_0^{\prime}(x) \prod_{j=2}^m f_j^{\prime}(x - P_j^{\prime}(y)) d\mu_{[N']}(y) d\mu(x), \end{align*} $$

with $P_j^{\prime }(y) = \ P_j(y_0 + y) - P_j(y_0)$ , $f_0^{\prime }(x) \ = \ f_0(x) \varphi _{N_1}*f_1(x - P_1(y_0))$ and $f_j^{\prime }(x) = f_j (x + P_j(y_0))$ . Note that each $f_j^{\prime }$ is supported in a fix dilate of I. In order to apply Theorem 6.1, we require $N' \simeq \delta ^M N \ge 1$ and here is where the condition $N \gtrsim \delta ^{-O_{\mathcal P}(1)}$ is needed. Therefore, Theorem 6.1 implies that

$$ \begin{align*}N_0^{-1} \|\mu_{[N_2]} * f_2 \|_{L^1(\mathbb{K})} \ = \ N_0^{-1} \|\mu_{[N_2]} * f_2^{\prime} \|_{L^1(\mathbb{K})} \ \gtrsim \ \delta^{O(1)}. \end{align*} $$

The equality of $L^1$ norms follows from the change of variables $x \to x + P_2(y_0)$ . This completes the proof of Equation (6.4) for $j=2$ when $\mathbb {K}$ is non-Archimedean since $\mu _{[N_2]} = \varphi _{N_2}$ .

We now turn to the Archimedian case, when $\mathbb {K} = {\mathbb R}$ or when $\mathbb {K}={\mathbb C}$ . Here, we argue as in Step 1 and establish the version of Equation (7.4) for the function $f_2$ . More precisely, writing

$$ \begin{align*}\Lambda_{{\mathcal P}; N}(f_0, \ldots, f_m) = \Lambda_{{\mathcal P};N}(f_0, f_1, \varphi_{N_2} * f_2, \ldots, f_m) + \Lambda_{{\mathcal P};N}(f_0, f_1, f_2 - \varphi_{N_2}*f_2, \ldots, f_m), \end{align*} $$

the argument in Step 1 shows that Equation (7.2) implies

(7.9) $$ \begin{align} |\Lambda_{{\mathcal P};N}(f_0, f_1, \varphi_{N_2} * f_2, \ldots, f_m)| \ \gtrsim \ \delta. \end{align} $$

This inequality allows us to reduce matters to showing that Equation (7.2) implies $N_0^{-1}\|\mu _{[N_2]}*f_2\|_{L^1(\mathbb {K})} \gtrsim \delta ^{O(1)}$ since then (7.9) would imply

$$ \begin{align*}\delta^{O(1)} \lesssim N_0^{-1} \|\mu_{[N_2]}*\varphi_{N_2}*f_2\|_{L^1(\mathbb{K})} \le N_0^{-1} \|\varphi_{N_2}*f_2\|_{L^1(\mathbb{K})}, \end{align*} $$

establishing (7.3) for $j=2$ .

We give the details when $\mathbb {K} = {\mathbb C}$ since there are additional technical difficulties alluded to above. The case ${\mathbb R}$ is easier. Given a large, general interval ${\mathcal I}$ in ${\mathbb C}$ (that is, ${\mathcal I}$ is a disc with large radius R), we can clearly find a mesh of $K\simeq \delta ^{-M}$ disjoint squares of side length $\delta ^{M/2} R$ which sit inside ${\mathcal I}$ such that . We fix such a mesh of squares for $[N]$ and a mesh of squares for $[N_0]$ so that

$$ \begin{align*}\Lambda_{{\mathcal P}, N}(f_0, \varphi_{N_1} * f_1, \ldots, f_m) \ = \end{align*} $$

Since $|\Lambda _{{\mathcal P}; N}(f_0, \varphi _{N_1}*f_1, \ldots , f_m)| \gtrsim \delta $ by Equation (7.4) and since the number of terms in each sum above is about $\delta ^{-M}$ , the pigeonhole principle gives us a square $T_0$ in $[N_0]$ and a square $S_0$ in $[N]$ such that

$$ \begin{align*}\Big| \frac{1}{N_0^{\prime} N'} \int_{T_0} \int_{S_0} f_0(x) f_0(x) \varphi_{N_1}*f_1(x - P_1(y)) \prod_{i=2}^m f_i(x - P_i(y)) d\mu(x) d\mu(y) \Big| \gtrsim \delta. \end{align*} $$

Write $[N']_{sq} = \{ z \in {\mathbb C}: |z |_{\infty } \le \sqrt {N'}\}$ , where $|z|_{\infty } = \max (|x|, |y|)$ for $z = x+iy$ . Hence, $S_0 = y_0 + [N']_{sq}$ for some $y_0 \in [N]$ . For $z \in S_0$ , we have $z = y_0 + y$ for some $y\in [N']_{sq}$ and so by the mean value theorem and the 1-boundedness of $f_1$ ,

$$ \begin{align*} &|\varphi_{N_1}* f_1 (x - P_1(y_0 + y)) - \varphi_{N_1}*f_1(x - P_1(y_0))| \\ &\le \ \sqrt{\frac{(N')^{\mathrm{Deg}P_1}}{N_1}} \int_{{\mathbb C}} \|(\nabla \varphi)_{N_1}(z)\| d\mu(z) \ \lesssim_{\varphi} \ \delta^{(M\mathrm{Deg}P_1 - C_1)/2}, \end{align*} $$

where $N_1 = \delta ^{C_1} N^{\mathrm {Deg}P_1}$ . Ensuring that $M \deg {P_1} - C_1 \ge 4$ , we see that

$$ \begin{align*}\Big| \frac{1}{N_0^{\prime} N'} \int_{T_0} \int_{[N']_{sq}} f_0(x) \varphi_{N_1}*f_1(x - P_1(y_0)) \prod_{i=2}^m f_i^t(x - P_i^{\prime}(y)) d\mu(x) d\mu(y) \Big| \gtrsim \delta, \end{align*} $$

where $P_i^{\prime }(y) = P_i(y_0 + y) - P_i(y_0)$ and $f_i^t(x) = f_i(x + P_i(y_0))$ . For an appropriate interval $I'$ containing $T_0$ with measure $\simeq N_0^{\prime }$ , we can write the above inequality as $|\Lambda _{{\mathcal P}'; N'}(f_0^{\prime }, f_2^{\prime }, \ldots , f_m^{\prime })| \gtrsim \delta $ , where ${\mathcal P}' = \{P_2^{\prime },\ldots , P_m^{\prime }\}$ , and for . Here,

$$ \begin{align*}\Lambda_{{\mathcal P}'; N'}(f_0^{\prime},\ldots, f_m^{\prime}) = \frac{1}{N_0^{\prime}} \iint_{{\mathbb C}^2} f_0^{\prime}(x) \prod_{i=2}^m f_i^{\prime}(x-P_i^{\prime}(y)) d\mu_{[N']_{sq}}(y) d\mu(x). \end{align*} $$

Again, in order to apply Theorem 6.1, we need $N' = \delta ^M N \ge 1$ which holds provided $N \gtrsim \delta ^{-O_{\mathcal P}(1)}$ . Therefore, by Theorem 6.1 (see the remark following the statement of Theorem 6.1), we conclude that

$$ \begin{align*} (N_0^{\prime})^{-1}\big\| \mu_{[N_2]_{sq}}*f_2^{\prime}\big\|_{L^1({\mathbb C})} \gtrsim_{\mathcal P} \delta^{O(1)} \end{align*} $$

for some $N_2\simeq \delta ^{C_2+M\deg (P_2)}N^{\deg (P_2)}$ . The function $\mu _{[N_2]}*f_2^{\prime }$ is supported on an interval $I"\supseteq I'$ such that $\mu (I"\setminus I') \lesssim N_2$ . Furthermore, we can find an interval $I"'\subseteq I'$ so that $\mu (I'\setminus I"') \lesssim N_2$ and for $x\in I"'$ , we have for all $u \in [N_2]_{sq}$ . Hence,

$$ \begin{align*}\delta^{O(1)} \lesssim \frac{1}{N_0^{\prime}} \int_{I"'} \Big| \int_{\mathbb C} f_2(x+P_2(y_0) - u) d\mu_{[N_2]_{sq}}(u) \Big| d\mu(x) \ + \ O(N_2 (N_0^{\prime})^{-1}), \end{align*} $$

where $N_2/N_0^{\prime } \lesssim \delta ^{M(\deg {P_2} - 1)}$ and $\deg {P_2} -1 \ge 1$ . Hence, for $M\gg 1$ sufficiently large, we conclude that

(7.10) $$ \begin{align} \delta^{O(1)} \lesssim \frac{1}{N_0^{\prime}} \int_{I"'} \Big| \int_{\mathbb C} f_2(x+P_2(y_0) - u) d\mu_{[N_2]_{sq}}(u) \Big| d\mu(x) \lesssim N_0^{-1} \|\mu_{[N_2]_{sq}} * f_2 \|_{L^1({\mathbb C})}. \end{align} $$

In the final inequality, we promoted the integration in x to all of ${\mathbb C}$ and changed variables $x \to x + P_2(y_0)$ . Hence, we have shown that Equation (7.2) implies $N_0^{-1}\|\mu _{[N_2]_{sq}}*f_2\|_{L^1({\mathbb C})} \gtrsim \delta ^{O(1)}$ . Since Equation (7.2) holds with $f_2$ replaced by $\varphi _{N_2}*f_2$ (this is Equation (7.9)), we see that

$$ \begin{align*}\delta^{O(1)} \lesssim N_0^{-1} \|\mu_{[N_2]_{sq}}*\varphi_{N_2}*f_2\|_{L^1({\mathbb C})} \le N_0^{-1} \|\varphi_{N_2}*f_2\|_{L^1({\mathbb C})}, \end{align*} $$

establishing Equation (7.3) for $j=2$ . Now, we can proceed inductively and obtain Equation (7.3) for all .

7.1 Multilinear functions and their duals

Recall the multilinear form

$$ \begin{align*}\Lambda_{{\mathcal P}; N}(f_0, f_1, \ldots, f_m) = \frac{1}{N_0} \iint_{\mathbb{K}^2} f_0(x) \prod_{i=1}^m f_i(x - P_i(y)) d\mu_{[N]}(y) d\mu(x). \end{align*} $$

We define the multilinear function

$$ \begin{align*} A_N^{\mathcal P}(f_1,\ldots, f_m)(x):=\int_{\mathbb{K}}\prod_{i=1}^mf_{i}(x-P_i(y))d\mu_{[N]}(y) \end{align*} $$

so that $\Lambda _{{\mathcal P}; N}$ can be written as a pairing of $A_N^{\mathcal P}$ with $f_0$ ,

$$ \begin{align*} \langle A_N^{\mathcal P}(f_1,\ldots, f_m), f_0\rangle = N_0 \, \Lambda_{{\mathcal P}; N, [N_0]}(f_0, f_1,\ldots, f_m), \end{align*} $$

where $\langle f, g\rangle = \int _{\mathbb {K}} f(x) g(x) d\mu (x)$ . By duality, we have

$$ \begin{align*} \langle A_N^{\mathcal P}(f_1,\ldots, f_m), f_0\rangle= \langle (A_N^{\mathcal P})^{*j}(f_1,\ldots, f_{j-1}, f_0, f_{j+1}, \ldots, f_m), f_j\rangle, \end{align*} $$

where

$$ \begin{align*} (A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m) (x) := \int_{\mathbb{K}}\prod_{\substack{i=1\\i\neq j}}^mf_{i}(x-P_i(y)+P_j(y))f_0(x+P_j(y))d\mu_{[N]}(y). \end{align*} $$

Lemma 7.11 (Application of Hahn–Banach).

Let $A,B> 0$ , let $I\subset \mathbb {K}$ be an interval, and let G be an element of $L^2(I)$ . Let $\Phi $ be a family of vectors in $L^2(I)$ , and assume the following inverse theorem: Whenever $f \in L^2(I)$ is such that $\|f\|_{L^\infty (I)} \leq 1$ and $|\langle f, G \rangle |> A$ , then $|\langle f, \phi \rangle |> B$ for some $\phi \in \Phi $ . Then G lies in the closed convex hull of

(7.12) $$ \begin{align} V= \{ \lambda\phi\in L^2(I): \phi \in \Phi, \ |\lambda| \leq A/B\} \cup \{ h \in L^2(I): \|h\|_{L^1(I)} \leq A \}. \end{align} $$

Proof. By way of contradiction, suppose that G does not lie in $W = \overline {\mathrm {conv} V}^{\|\cdot \|_{L^2(I)}}$ . From the Hahn–Banach theorem, we can find a continuous linear functional $\Lambda $ of $L^2(I)$ which separates G from W; that is, there is a $C \in {\mathbb R}$ such that $\operatorname {\mathrm {Re}}\Lambda (h) \le C < \operatorname {\mathrm {Re}} \Lambda (G)$ for all $h \in W$ . Scaling $\Lambda $ allows us to change the constant C, so we can choose $\Lambda $ such that $C=A$ is in the statement of the lemma. Since W is balanced, we see that $|\Lambda (h)| \le A < \operatorname {\mathrm {Re}} \Lambda (G)$ for all $h \in W$ . By the Riesz representation theorem, there is an $f \in L^2(I)$ which represents $\Lambda $ so that $|\langle f, h\rangle |\le A < \mathrm {Re} \langle f, G \rangle $ for all $h\in V$ . This implies that

$$\begin{align*}|\langle f, \phi\rangle| \leq B \end{align*}$$

for all $\phi \in \Phi $ and that

$$\begin{align*}\|f\|_{L^\infty(I)}=\sup_{\|h\|_{L^1(I)}\le1} |\langle f, h\rangle|\le 1, \end{align*}$$

contradicting the hypothesis of the lemma. This completes the proof of the lemma.

Corollary 7.13 (Structure of dual functions).

Let $N\ge 1$ be a scale, $m\in \mathbb Z_+$ and $0<\delta \le 1$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of polynomials such that $1\le \deg {P_1}<\ldots <\deg {P_m}$ . Let $f_0, f_1,\ldots , f_m\in L^0(\mathbb {K})$ be $1$ -bounded functions supported on an interval of measure $N_0 = N^{\deg (P_m)}$ . Then for every , provided $N\gtrsim \delta ^{-O_{\mathcal P}(1)}$ , there exist a decomposition

(7.14) $$ \begin{align} (A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)(x) = H_j(x) + E_j(x), \end{align} $$

where $H_j \in L^2(\mathbb {K})$ has Fourier transform supported in $[(N_j)^{-1}]$ , where $N_j \simeq \delta ^{C_j} N^{\deg {P_j}}$ and $C_j$ is as in Theorem 7.1, and obeys the bounds

(7.15) $$ \begin{align} \|H_j\|_{L^\infty(\mathbb{K})} \lesssim_m 1, \quad\text{ and }\quad \|H_j\|_{L^1(\mathbb{K})} \lesssim_m N_0. \end{align} $$

The error term $E_j\in L^1(\mathbb {K})$ obeys the bound

(7.16) $$ \begin{align} \|E_j\|_{L^1(\mathbb{K})} \leq \delta N_0. \end{align} $$

Proof. Fix , let $I_0:=\operatorname {\mathrm {supp}}{(A_N^{\mathcal P})^{*j}(f_1,\ldots , f_0, \ldots , f_m)}$ and recall that $N_0= N^{\deg (P_m)}$ . By translation invariance we may assume $\operatorname {\mathrm {supp}}{f_j}\subseteq [N_0]$ for all , and that $I_0:=[O(N_0)]$ . If there exists $f\in L^{\infty }(I_0)$ with $\|f\|_{L^{\infty }(I_0)}\le 1$ such that

(7.17) $$ \begin{align} |\langle f, (A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m) \rangle|> \delta N_0, \end{align} $$

then proceeding as in the proof of Theorem 7.1 we may conclude that

$$ \begin{align*} |\langle \varphi_{N_j}*f, (A_N^{\mathcal P})^{*j}(\varphi_{N_1}*f_1,\ldots, \varphi_{N_{j-1}}* f_{j-1}, f_0, f_{j+1} \ldots, f_m) \rangle|\ge c_m \, \delta N_0, \end{align*} $$

where $N_i\simeq \delta ^{C_i} N^{\deg (P_i)}$ for . This implies that there exists a 1-bounded $F\in L^2(\mathbb {K})$ with $\|F\|_{L^1(\mathbb {K})} \le N_0$ such that $\operatorname {\mathrm {supp}}{\widehat {F}}\subseteq [N_j^{-1}]$ and

(7.18) $$ \begin{align} |\langle f, F \rangle|\ge c_m \, \delta N_0. \end{align} $$

If fact, we can take

$$ \begin{align*}F(x) = {\tilde{\varphi}}_{N_j}*(A_N^{\mathcal P})^{*j}(\varphi_{N_1}*f_1,\ldots, \varphi_{N_{j-1}}* f_{j-1}, f_0, f_{j+1} \ldots, f_m)(x), \end{align*} $$

where ${\tilde {\varphi }}(x) = \varphi (-x)$ . Let $\Psi $ denote the collection of all 1-bounded $F\in L^2(\mathbb {K})$ with $\operatorname {\mathrm {supp}}{\widehat {F}}\subseteq [N_j^{-1}]$ and $\|F\|_{L^1(\mathbb {K})}\le N_0$ . Invoking Lemma 7.11 with $A=\delta N_0/4$ and $B = c_m \delta N_0$ and the set , we obtain a decomposition

(7.19) $$ \begin{align} (A_N^{\mathcal P})^{*j}(f_1,\ldots, f_{j-1}, f_0, f_{j+1}, \ldots, f_m) = \sum_{l=1}^{\infty}c_l\phi_l+E(1)+E(2), \end{align} $$

with the following properties:

1. (i) for each $l\in \mathbb Z_+$ , we have that , $F_l \in \Psi $ and $\lambda _l\in \mathbb C$ such that $|\lambda _l|\lesssim _m 1$ ;

2. (ii) the coefficients $c_l$ are nonnegative with $\sum _{l=1}^{\infty }c_l\le 1$ , and all but finitely $c_l$ vanish;

3. (iii) the error term $E(1)\in L^1(I_0)$ satisfies $\|E(1)\|_{L^1(I_0)} \leq \delta N_0/2$ ;

4. (iv) the error term $E(2)\in L^2(I_0)$ satisfies $\|E(2)\|_{L^2(I_0)} \leq \delta $ .

The latter error term arises as a consequence of the fact that one is working with the closed convex hull instead of the convex hull. In fact, its $L^2(I_0)$ norm can be made arbitrarily small, but $\delta $ will suffice for our purposes.

Grouping together terms in the decomposition (7.19), we have

$$\begin{align*}(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_{j-1}, f_0, f_{j+1}, \ldots, f_m) = H_j^{\prime} + E_j^{\prime}, \end{align*}$$

where

$$ \begin{align*} \|H_j^{\prime}\|_{L^{\infty}(\mathbb{K})} \le \sup_{l\in\mathbb N} \|F_l\|_{L^{\infty}(\mathbb{K})} \sum_{l=1}^{\infty} c_l |\lambda_l| \lesssim_m 1. \end{align*} $$

Also, $E_j^{\prime } = E(1) + E(2)$ satisfies $\|E_j^{\prime }\|_{L^1(I_0)} \le \delta N_0$ by (iii) and (iv) above since by the Cauchy–Schwarz inequality, we have $\|E(2)\|_{L^1(I_0)} \leq \delta N_0^{1/2}$ .

We note that the function $F(x) = \sum _{i=1}^{\infty } c_i \lambda _i F_i (x)$ is Fourier supported in the interval $[N_j^{-1}]$ .

When $\mathbb {K}$ is non-Archimedean, and so the Fourier transform of $H_j^{\prime }$ is supported in $[N_j^{-1}]$ . This verifies Equation (7.15) in this case and completes the proof when $\mathbb {K}$ is non-Archimedean since the decomposition $H_j^{\prime } + E_j^{\prime }$ of $(A_N^{\mathcal P})^{*j}$ satisfies Equations (7.15) and (7.16).

Now, suppose $\mathbb {K}$ is Archimedean. Let $\psi $ be a Schwartz function such that $\int _{\mathbb {K}}\psi (x)d\mu (x)=1$ and $\operatorname {\mathrm {supp}}{\widehat {\psi }}\subseteq [2]$ . Let $M\simeq \delta ^{O(1)} N_0$ and as usual, set $\psi _M(x) = M^{-1} \psi (M^{-1}x)$ when $\mathbb {K} = {\mathbb R}$ and $\psi _M(x) = M^{-1}\psi (M^{-1/2} x)$ when $\mathbb {K} = {\mathbb C}$ . From the proof of Equation (7.8), we have

(7.20)

We set and so that

$$ \begin{align*}(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_{j-1}, f_0, f_{j+1}, \ldots, f_m) (x) = H_j(x) + E_j (x). \end{align*} $$

From Equation (7.20), we see that $E_j$ satisfies Equation (7.16). The properties $\|H_j\|_{L^\infty (\mathbb {K})} \lesssim _m 1$ and $\|H_j\|_{L^1(\mathbb {K})} \lesssim _m N_0$ are still preserved. Moreover, $\operatorname {\mathrm {supp}}{\widehat {H}_j}\subseteq [O(N_j^{-1})]$ since

The shows that Equation (7.15) holds for $H_j$ and this completes the proof of the corollary.

We will combine Corollary 7.13 and the following $L^p$ improving bound for polynomial averages to establish the key Sobolev inequality.

Lemma 7.21 ( $L^p$ -improving for polynomial averages).

Let $Q \in \mathbb {K}[\mathrm {y}]$ with $\deg (Q) = d$ , and let $N\gg _Q 1$ be a large scale. Consider the averaging operator

$$\begin{align*}M_N^{Q}g(x):=\int_{\mathbb{K}} g(x-Q(y))d\mu_{[N]}(y). \end{align*}$$

For any parameters $1 < r < s< \infty $ satisfying $1/s = 1/r - 1/d$ , the following inequality holds:

(7.22) $$ \begin{align} \|M_N^{Q}g\|_{L^s(\mathbb{K})}\lesssim_Q N^{d(\frac{1}{s}-\frac{1}{r})}\|g\|_{L^r(\mathbb{K})}\quad \text{ for } \quad g\in L^r(\mathbb{K}). \end{align} $$

Proof. As our bounds are allowed to depend on Q, we may assume that Q is monic. Let $\alpha \in \mathbb {K}$ be such that $|\alpha | = N$ , and change variables $y \to \alpha y$ to write

$$ \begin{align*}M^N_Q g(x) \ = \int_{B_1(0)} g(x - Q(\alpha y)) \, d\mu(y) = \int_{B_1(0)} g_{\alpha}(\alpha^{-d} x - Q_{\alpha}(y)) \, d\mu(y) \end{align*} $$

where $g_{\alpha }(x) = g(\alpha ^d x)$ and $Q_{\alpha }(y) = \alpha ^{-d} Q(\alpha y) = y^d + \alpha ^{-1} a_{d-1} y^{d-1} + \ldots +\alpha ^{-d}a_0$ . Hence, the right-hand side above can be written as $M^1_{Q_{\alpha }} g_{\alpha }( \alpha ^{-d} x)$ . Since $\|g_{\alpha }\|_{L^r(\mathbb {K})} = N^{-d/r} \|g\|_{L^r(\mathbb {K})}$ , we see that matters are reduced to proving Equation (7.22) for $N=1$ and $Q = Q_{\alpha }$ with uniform bounds in $\alpha $ .

The mapping $y \to Q_{\alpha }(y)$ is d-to- $1$ , and we can use a generalised change of variables formula to see that

$$ \begin{align*} |M^1_{Q_{\alpha}} g(x)| \lesssim \int_{|s|\le 2} |g(x -s)| |s|^{-(d-1)/d} d\mu(s) \end{align*} $$

when $N\gg _{Q} 1$ . Hence $M^1_{\alpha }$ is controlled by fractional integration, uniformly in $\alpha $ . When $\mathbb {K}$ is Archimedean, such a change of variables formula is well known. Recall that when $\mathbb {K} = {\mathbb C}$ , $|s| = s {\overline {s}}$ is the square of the usual absolute value.

When $\mathbb {K} = {\mathbb Q}_p$ is the p-adic field, such a formula is given in [12]. The argument in [12] generalises to general non-Archimedean fields (when the characteristic, if positive, is larger than d). Alternatively, one can use a construction in [46], valid in any local field and valid for any polynomial Q where $Q'(x)$ does not equal to zero mod $m_{\mathbb {K}}$ for any nonzero x (we need the condition on the characteristic of the field for this), in which the unit group is partitioned into $J=\mathrm {gcd}(d, q-1)$ open sets and analytic isomorphisms $\phi _j :D_j \to \phi _j(D_j)$ are constructed such that $y = \phi _j(x)$ precisely when $Q(y) = x$ . For us, $Q_{\alpha }'(x) \not = 0$ mod $m_{\mathbb {K}}$ for any nonzero x if $|\alpha | = N \gg _Q 1$ is sufficiently large.

By the Hardy–Littlewood–Sobolev inequality (easily seen to be valid over general locally compact topological fields), we have

$$ \begin{align*}\|M^1_{Q_{\alpha}} g \|_{L^s(\mathbb{K})} \lesssim \|g\|_{L^r(\mathbb{K})}, \end{align*} $$

uniformly in $\alpha $ whenever $1/s = 1/r - 1/d$ , completing the proof of the lemma.

We now come to the proof of Theorem 1.6.

As in the set up for Theorem 7.1, we fix a smooth function $\varphi $ with compact Fourier support. When $\mathbb {K}$ is Archimedean, let $\varphi $ be a Schwartz function on $\mathbb {K}$ so that

When $\mathbb {K}={\mathbb R}$ , we set $\varphi _N(x) = N^{-1} \varphi (N^{-1} x)$ for any $N>0$ and when $\mathbb {K} = {\mathbb C}$ , we set $\varphi _{N}(z) = N^{-1} \varphi (N^{-1/2} z)$ for any $N> 0$ . When $\mathbb {K}$ is non-Archimedean, we set so that and we set for any scale N. We restate Theorem 1.6 in a more formal, precise way.

Theorem 7.23 (A Sobolev inequality for $A_N^{\mathcal P}$ ).

Let ${\mathcal P}:=\{P_1,\ldots , P_m\}$ be a collection of polynomials such that $1\le \deg {P_1}<\ldots <\deg {P_m}$ . Let $N \gg _{\mathcal P} 1$ be a scale, $m\in \mathbb Z_+$ and $0<\delta \le 1$ be given. Let $1<p_1,\ldots , p_m<\infty $ satisfying $\frac {1}{p_1}+\ldots +\frac {1}{p_m}=1$ be given. Suppose $N\gtrsim \delta ^{-O_{\mathcal P}(1)}$ . Then for all $f_1\in L^{p_1}(\mathbb {K}),\ldots , f_m\in L^{p_m}(\mathbb {K})$ we have

(7.24) $$ \begin{align} \|A_N^{\mathcal P}(f_1,\ldots,f_{j-1}, (\delta_0-\varphi_{N_j})*f_j,f_{j+1}\ldots, f_m)\|_{L^1(\mathbb{K})} \lesssim \delta^{1/8} \prod_{i=1}^{m} \|f_i\|_{L^{p_i}(\mathbb{K})}, \end{align} $$

where $N_j \simeq \delta ^{C_j} N^{\deg {P_j}}$ and $C_j$ is the parameter from Theorem 7.1. Here, $\widehat {\delta _0} \equiv 1$ .

Remark. The proof of Theorem 7.23 (and its statement) implicitly assumes that $m\ge 2$ , but there is a version when $m=1$ , which will be given in Section 8 where it is needed.

Proof. We fix and recall $N_j\simeq \delta ^{O(1)}N^{\deg (P_j)}$ . We first prove that for every functions $f_1,\ldots , f_{j-1}, f_{j+1},\ldots , f_{m-1}\in L^{\infty }(\mathbb {K})$ and $f_j, f_m\in L^2(\mathbb {K})$ , we have

(7.25) $$ \begin{align} \begin{split} \|A_N^{\mathcal P}(f_1,\ldots,f_{j-1}, (\delta_0-\varphi_{N_j})*f_j,f_{j+1}\ldots, f_m)\|_{L^1(\mathbb{K})}\\ \lesssim \delta^{1/8} \bigg(\prod_{\substack{i=1\\i\neq j}}^{m-1} \|f_i\|_{L^\infty(\mathbb{K})}\bigg)\|f_j\|_{L^2(\mathbb{K})}\|f_m\|_{L^2(\mathbb{K})}.\qquad \end{split} \end{align} $$

Choose $f_0\in L^{\infty }(\mathbb {K})$ so that $\|f_0\|_{L^{\infty }(\mathbb {K})}=1$ and

$$ \begin{align*} \|A_N^{\mathcal P}(f_1,&\ldots,f_{j-1}, (\delta_0-\varphi_{N_j})*f_j,f_{j+1}\ldots, f_m)\|_{L^1(\mathbb{K})}\\ &\simeq |\langle A_N^{\mathcal P}(f_1,\ldots,f_{j-1}, (\delta_0-\varphi_{N_j})*f_j,f_{j+1}\ldots, f_m), f_0\rangle|\\ &=|\langle (\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m), f_j\rangle|. \end{align*} $$

By the Cauchy–Schwarz inequality, it will suffice to prove

(7.26) $$ \begin{align} \begin{split} \|(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)\|_{L^2(\mathbb{K})}\\ \lesssim \delta^{1/8} \|f_0\|_{L^\infty(\mathbb{K})}\bigg(\prod_{\substack{i=1\\i\neq j}}^{m-1} \|f_i\|_{L^\infty(\mathbb{K})}\bigg)\|f_m\|_{L^2(\mathbb{K})}. \end{split} \end{align} $$

By multilinear interpolation, the bounds (7.25) imply Equation (7.24) and so the proof of Theorem 7.23 is reduced to establishing Equation (7.26) which will be divided into three steps. In the first two steps, we will assume that $f_m$ is supported in some interval of measure $N_0$ where $N_0 \simeq N^{\deg (P_m)}$ .

Step 1

In this step, we will establish the bound

(7.27) $$ \begin{align} \begin{split} \|(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)\|_{L^2(\mathbb{K})}\quad \\ \lesssim \delta^{1/2}N_0^{1/2} \|f_0\|_{L^\infty(\mathbb{K})}\bigg(\prod_{\substack{i=1\\ i\neq j}}^{m-1} \|f_i\|_{L^\infty(\mathbb{K})}\bigg)\|f_m\|_{L^\infty(\mathbb{K})} \end{split} \end{align} $$

under the assumption that $f_m$ is supported in an interval of measure $N_0$ (when $\mathbb {K} = {\mathbb C}$ , this implies in particular that $f_m$ is supported in a square with measure about $N_0$ , which in Step 3 will be a helpful observation). When $f_m$ has this support condition,

$$ \begin{align*}(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m) = (A_N^{\mathcal P})^{*j}(f_1^{\prime},\ldots, f_0^{\prime}, \ldots, f_m^{\prime}), \end{align*} $$

where for some interval $I_0$ of measure $O(N_0)$ . To prove Equation (7.27), it suffices to assume $\|f_i\|_{L^{\infty }(\mathbb {K})} = 1$ for $i=0,1,\ldots , j-1, j+1,\ldots , m$ and so Equation (7.27) takes the form

(7.28) $$ \begin{align} \|(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)\|_{L^2(\mathbb{K})}\lesssim \delta^{1/2}N_0^{1/2}. \end{align} $$

We apply the decomposition (7.14) to $(A_N^{\mathcal P})^{*j}(f_1^{\prime },\ldots , f_0^{\prime }, \ldots , f_m^{\prime })$ to write

$$ \begin{align*}(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m) (x) = H_j(x) + E_j(x), \end{align*} $$

where $H_j$ satisfies Equation (7.15) and $E_j$ satisfies Equation (7.16). Using the fact that $\widehat {H}_j\subseteq [(N_j)^{-1}]$ , we conclude that $(\delta _0-\varphi _{N_j})*H_j=0$ . Thus,

$$\begin{align*}(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)=(\delta_0-\varphi_{N_j})*E_j. \end{align*}$$

From Equation (7.16) and the 1-boundedness of $(A_N^{\mathcal P})^{*j}(f_1,\ldots , f_0, \ldots , f_m)$ , we have

$$ \begin{align*} \|(\delta_0 -\varphi_{N_j})*E_j\|_{L^1(\mathbb{K})}\lesssim \delta N_0, \quad \text{ and } \quad \|(\delta_0 -\varphi_{N_j})*E_j\|_{L^\infty(\mathbb{K})}\lesssim 1, \end{align*} $$

respectively. Therefore,

$$ \begin{align*} \|(\delta_0 -\varphi_{N_j})*E_j\|_{L^2(\mathbb{K})}\lesssim \delta^{1/2} N_0^{1/2}, \end{align*} $$

establishing (7.28) and hence Equation (7.27). This completes Step 1.

Step 2.

We continue with our assumption that $f_m$ is supported in an interval of measure $N_0$ , but now we relax the $L^\infty (\mathbb {K})$ control on $f_m$ to $L^2(\mathbb {K})$ control and show that

(7.29) $$ \begin{align} \begin{split} \|(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)\|_{L^2(\mathbb{K})}\\ \lesssim \delta^{1/4}\|f_0\|_{L^\infty(\mathbb{K})}\bigg(\prod_{\substack{i=1\\i\neq j}}^{m-1} \|f_i\|_{L^\infty(\mathbb{K})}\bigg)\|f_m\|_{L^2(\mathbb{K})}. \end{split} \end{align} $$

The main tool for this will be the $L^p$ -improving estimate (7.22) for the polynomial average $M_N^Q$ . We have a pointwise bound

$$ \begin{align*} |(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)(x)|\le M_N^{P_m-P_j}|f_m|(x), \end{align*} $$

which combined with Equation (7.22) (for $Q=P_m-P_j$ , $d = \deg (P_m)$ , $s=2$ and $r=(d+2)/2d$ ) yields

(7.30) $$ \begin{align} \begin{split} \|(\delta_0-\varphi_{N_j})*(A_N^{\mathcal P})^{*j}(f_1,\ldots, f_0, \ldots, f_m)\|_{L^2(\mathbb{K})}\quad\\ \lesssim N_0^{- 1/d}\|f_0\|_{L^\infty(\mathbb{K})}\bigg(\prod_{\substack{i=1\\i\neq j}}^{m-1} \|f_i\|_{L^\infty(\mathbb{K})}\bigg)\|f_m\|_{L^r(\mathbb{K})}. \end{split} \end{align} $$

Interpolating Equations (7.27) and (7.30), we obtain Equation (7.29) as desired.

Step 3.

In this final step, we remove the support condition on $f_m$ and establish (7.26). To prove Equation (7.26), we may assume that $\|f_i\|_{L^{\infty }(\mathbb {K})} = 1$ for $i=0, 1, \ldots , j-1, j+1, \ldots , m-1$ . We split , where I ranges over a partition $\mathcal I$ of $\mathbb {K}$ into intervals I of measure $N_0$ . We have seen this is possible when $\mathbb {K}$ is non-Archimedean or when $\mathbb {K}={\mathbb R}$ . This is not possible when $\mathbb {K} = {\mathbb C}$ , but in this case, we can find a partition $\mathcal I$ of squares. By Step 1 and Step 2, the local dual function obeys the bound

(7.31) $$ \begin{align} \| (\delta_0 - \varphi_{N_j})* D_I \|_{L^2(\mathbb{K})} \lesssim \delta^{1/4} \| f_m \|_{L^2(I)} \end{align} $$

for each interval I, and we wish to establish

$$\begin{align*}\Big\| \sum_{I\in\mathcal I} (\delta_0 - \varphi_{N_j})* D_I\Big\|_{L^2(\mathbb{K})} \lesssim\delta^{1/8} \| f_m \|_{L^2(\mathbb{K})}. \end{align*}$$

We will square out the sum. To handle the off-diagonal terms, we observe that for finite intervals $I, J\subset \mathbb {K}$ (squares when $\mathbb {K} = {\mathbb C}$ ) of measure $N_0$ and $M>0$ and $1\le p<\infty $ , we have

(7.32)

By squaring and applying Schur’s test, it suffices to obtain the decay bound

$$\begin{align*}\bigl| \langle (\delta_0 - \varphi_{N_j})* D_I, (1 - \varphi_{N_j})* D_J \rangle \bigr| \lesssim \delta^{1/4} \big(1+N_0^{-1}\mathrm{dist}(I,J) \big)^{-2} \| f_m \|_{L^2(I)} \| f_m \|_{L^2(J)} \end{align*}$$

for all intervals $I,J$ of measure $N_0$ . By Cauchy–Schwarz and Equation (7.31), we know

$$\begin{align*}\langle (\delta_0 - \varphi_{N_j})* D_I, (1 - \varphi_{N_j})* D_J \rangle \lesssim \delta^{1/2} \| f_m \|_{L^2(I)} \| f_m \|_{L^2(J)}. \end{align*}$$

On the other hand, $D_I$ is supported in a $O(N_0)$ -neighborhood of I, and similarly for $D_J$ . From Equation (7.32) and Cauchy–Schwarz, we thus have

$$ \begin{align*} \langle (\delta_0 - \varphi_{N_j})* D_I, (1 - \varphi_{N_j})* D_J \rangle &\lesssim \big(1+N_0^{-1}\mathrm{dist}(I,J) \big)^{-10} \| D_I \|_{L^2(\mathbb{K})} \| D_J \|_{L^2(\mathbb{K})}\\ &\lesssim \big(1+N_0^{-1}\mathrm{dist}(I,J) \big)^{-10} \| f_m \|_{L^2(I)} \| f_m \|_{L^2(J)}. \end{align*} $$

Taking the geometric mean of the two estimates, we obtain the claim in Equation (7.26). This completes the proof of Theorem 7.23.

8 The implication Theorem 1.6  $\Longrightarrow $ Theorem 1.3

Here, we give the details of Bourgain’s argument in [3] which allow us to pass from Theorem 1.6 to Theorem 1.3 on polynomial progressions. Let ${\mathcal P} = \{P_1, \ldots , P_m\}$ be a sequence of polynomials in $\mathbb {K}[\mathrm {y}]$ with distinct degrees and no constant terms. Without loss of generality, we may assume

$$ \begin{align*}\deg{P_1} < \deg{P_2} < \cdots < \deg{P_m}, \end{align*} $$

and we set $d_{m-j} := \deg {P_j}$ and $d := d_0 = \deg {P_m}$ so that $d_{m-1} < \cdots < d_1 < d$ .

Since the argument showing how Theorem 1.6 implies Theorem 1.3 has been given in [3], [11] and [8] in the Euclidean setting (albeit for shorter polynomial progressions), we will only give the details for non-Archimedean fields $\mathbb {K}$ where uniform notation can be employed.

We will proceed in several steps.

Step 1

When $\mathbb {K}$ is non-Archimedean, the family $(Q_t)_{t>0}$ of convolution operators defined by

$$\begin{align*}Q_t f (x) \ = \ f * \mu_{[t]}(x) = \ \frac{1}{t} \int_{|y|\le t} f(x - u) d\mu(u) \quad \text{for scales} \quad t>0 \end{align*}$$

gives us a natural approximation of the identity and form the analogue of the Poisson semigroup in the non-Archimedean setting. They also give us Fourier localization since

(8.1)

We will need the following bound for $(Q_t)_{t>0}$ (see Lemma 6 in [3] or Lemma 2.1 in [11]): For $f \ge 0$ and scales $0 < t_1, \ldots , t_m \le 1$ ,

(8.2) $$ \begin{align} \int_{B_1(0)} f(x) Q_{t_1} f(x) \cdots Q_{t_m} f(x) d\mu(x) \ \ge \ \Big( \int_{B_1(0)} f(x) d\mu(x) \Big)^{m+1}. \end{align} $$

The proof in the euclidean setting given in [11] established Equation (8.2) for general approximations of the identity, but the first step is to show Equation (8.2) for martingales $(E_k)_{k\in \mathbb {N}}$ defined with respect to dyadic intervals. However, a small scale t in a non-Archimedean field $\mathbb {K}$ is the form $t = q^{-k}$ and

$$\begin{align*}{\mathcal C}_k \ = \ \{ {\underline{x}}=x_0 + x_1 \pi + \cdots + x_{k-1}\pi^{k-1} : x_j \in o_{\mathbb{K}}/m_{\mathbb{K}} \} \quad \text{ and } \quad A_{k,\underline{x}}f = q^{k} \int_{B_{q^{-k}({\underline{x}})}} f(u) d\mu(u). \end{align*}$$

Hence, $(Q_t)_{t>0}$ is a martingale with respect to the dyadic structure of non-Archimedean fields and so the argument in [11] extends without change to establish Equation(8.2).

Step 2

Fix $\varepsilon>0$ . Our goal is to find a $\delta (\varepsilon , {\mathcal P})> 0$ and $N(\varepsilon , {\mathcal P}) \ge 1$ such that for any scale $N \ge N(\varepsilon , {\mathcal P})$ and $f \in L^0(\mathbb {K})$ with $0\le f \le 1$ satisfying $\int _{\mathbb {K}} f d\mu \ge \varepsilon N^d$ , we have

(8.3) $$ \begin{align} I := \frac{1}{N^d} \iint_{\mathbb{K}^2} f(x) f(x+P_1(y)) \cdots f(x+P_m(y)) d\mu_{[N]}(y) d\mu(x) \ \ge \ \delta. \end{align} $$

Taking with $S\subseteq \mathbb {K}$ in Theorem 1.3 implies Equation (1.4), the desired conclusion. We may assume the f is supported in the interval $[N^d]$ .

Let $\alpha , \beta \in \mathbb {K}$ satisfy $|\alpha | = N^d$ and $|\beta | = N$ , and write

$$ \begin{align*}I = \iint_{\mathbb{K}^2} g(x) g(x + R_1(y)) \cdots g(x +R_m(y)) d\mu_{[1]}(y) d\mu(x), \end{align*} $$

where $g(x) = f(\alpha x)$ and $R_j(y) = \alpha ^{-1} P_j(\beta y)$ . In particular, we have $\int _{\mathbb {K}} g \ge \varepsilon $ . We note that g is supported in $[1] = B_1(0)$ . Fix three small scales $0<t_0 \ll t_1 \ll t \ll 1$ and decompose

(8.4) $$ \begin{align} t_1^{-1} I \ge \iint_{\mathbb{K}^2} g(x) g(x + R_1(y)) \cdots g(x + R_m(y)) d\mu_{[t_1]}(y) d\mu(y) \ =: \ I_1 + I_2 + I_3, \end{align} $$

where

$$ \begin{align*}I_1 &= \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-1} g(x + R_j(y))\ Q_t g(x +R_m(y)) d\mu_{[t_1]}(y) d\mu(x),\\I_2 &= \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-1} g(x + R_j(y))\ [Q_{t_0} - Q_t ] g(x +R_m(y)) d\mu_{[t_1]}(y) d\mu(x) \ \ \mathrm{and}\\I_3 &= \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-1} g(x + R_j(y))\ [ \mathrm{Id} - Q_{t_0}] g(x +R_m(y)) d\mu_{[t_1]}(y) d\mu(x). \end{align*} $$

For $I_1$ , we note that for $t_1 \ll _{P_m} t$ ,

$$ \begin{align*}Q_t g(x + R_m(y)) = \frac{1}{t} \int_{|u|\le t} g(x + R_m(y) - u) d\mu(u) = \frac{1}{t} \int_{|u|\le t} g(x-u) d\mu(u) = Q_t g(x) \end{align*} $$

whenever $|y| \le t_1$ . For the final equality, we made the change of variables $u \to u - R_m(y)$ , noting that when $|y| \le t_1$ , then $|R_m(y)| \le C_{P_m} t_1 \le t$ . Hence,

$$ \begin{align*} I_1 = \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-1} g(x + R_j(y))\ Q_t g(x)\, d\mu_{[t_1]}(y) d\mu(x). \end{align*} $$

For $I_2$ , we use the Cauchy–Schwarz inequality to see that

(8.5) $$ \begin{align} I_2 \ \le \ \| Q_{t_0} g - Q_t g \|_{L^2(\mathbb{K})}. \end{align} $$

For $I_3$ , we will use the more precise formulation of Theorem 1.6 given in Theorem 7.23. We rescale $I_3$ , moving from $g, R_j$ back to $f, P_j$ and write

$$\begin{align*}I_3 = \frac{1}{N^d} \iint_{\mathbb{K}^2} f(x) \prod_{j=1}^{m-1} f(x + P_j(y))\ [\mathrm{Id} - Q_{t_0 N^d}] f (x + P_m(y)) d\mu_{[t_1N]}(y) d\mu(x), \end{align*}$$

where the function $h(x) = [\mathrm {Id} - Q_{t_0N^d}] f(x)$ has the property that ${\widehat {h}}(\xi ) = 0$ whenever $|\xi | \le (t_0 N^d)^{-1}$ ; See (8.1). Hence,

$$ \begin{align*}I_3 \le N^{-d} \|A^{\mathcal P}_{t_1N}(f,f, \ldots, f, [\mathrm{Id} - Q_{t_0N^d}] f) \|_{L^1(\mathbb{K})}, \end{align*} $$

and we will want to apply Theorem 7.23 to the expression on the right with N replaced by $t_1N$ and $0 < \delta \le 1$ defined by $\delta ^{C_m} (N t_1)^d = N^d t_0$ or $\delta = (t_0/ t_1^d)^{1/C_m}$ . In order to apply Theorem 7.23, we will need to ensure

(8.6) $$ \begin{align} N \ \ge \ t_1^{-1} (t_1^{d_{m-1}} / t_0)^{C'} \ge\ldots \ge \ t_1^{-1} (t_1^{d} / t_0)^{C'} \end{align} $$

for some appropriate large $C' = C^{\prime }_{\mathcal P}$ . If Equation (8.6) holds, then Theorem 7.23 implies there exists a constant $b = b_{\mathcal P}> 0$ such that

$$ \begin{align*}\|A^{\mathcal P}_{t_1N}(f,f, \ldots, f, h)\|_{L^1(\mathbb{K})} \lesssim_{\mathcal P} \bigl(t_0/t_1^d\bigr)^b \prod_{j=1}^m \|f\|_{L^{p_i}(\mathbb{K})} \le \bigl(t_0/t_1^d\bigr)^b N^d \end{align*} $$

since $1/p_1 + \cdots + 1/p_m = 1$ and $\|f\|_{L^{p_i}(\mathbb {K})} \le N^{d/p_i}$ for (which follows since f is 1-bounded and supported in $[N^d]$ ). Hence,

$$ \begin{align*} I_3 \lesssim_{\mathcal P} \bigl(t_0/t_1^d\bigr)^b \quad \text{ if } \quad ({8.6})\quad \text{holds}. \end{align*} $$

Step 3

Next we decompose $I_1 = I_1^1 + I_{2}^1 + I_3^1$ , where

$$ \begin{align*}I_1^1 \ = \ \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-2} g(x + R_j(y))\ Q_{t/N^{d-d_1}} g(x + R_{m-1}(y)) Q_t g(x) \, d\mu_{[t_1]}(y) d\mu(x), \end{align*} $$

$$ \begin{align*}I_{2}^1 = \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-2} g(x + R_j(y))\ [Q_{t_0/N^{d-d_1}} - Q_{t/N^{d-d_1}} ] g(x +R_{m-1}(y)) Q_t g(x) d\mu_{[t_1]}(y) d\mu(x) \ \ \mathrm{and} \end{align*} $$

$$ \begin{align*}I_{3}^1 = \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-2} g(x + R_j(y))\ [ \mathrm{Id} - Q_{t_0/N^{d-d_1}}] g(x +R_{m-1}(y)) Q_t g(x) d\mu_{[t_1]}(y) d\mu(x). \end{align*} $$

For $I_1^1$ , we set $s = t/N^{d-d_1}$ and note that for $t_1 \ll _{\mathcal P} t$ ,

$$ \begin{align*}Q_{s} g(x + R_{m-1}(y)) = \frac{1}{s} \int_{|u|\le s} g(x + R_{m-1}(y) - u) d\mu(u) = \frac{1}{s} \int_{|u|\le s} g(x-u) d\mu(u) = Q_s g(x) \end{align*} $$

whenever $|y| \le t_1$ . For the final equality we made the change of variables $u \to u - R_{m-1}(y)$ , noting that when $|y| \le t_1$ , then $|R_{m-1}(y)| \le C_{P_{m-1}} N^{-(d-d_1)}t_1 \le s$ since $ t_1 \ll _{\mathcal P} t$ . Hence,

$$ \begin{align*} I_1^1 = \iint_{\mathbb{K}^2} g(x) \prod_{j=1}^{m-2} g(x + R_j(y))\ Q_{t/N^{d-d_1}} g(x) Q_t g(x) \, d\mu_{[t_1]}(y) d\mu(x). \end{align*} $$

As in Equation (8.5), we have

$$ \begin{align*}I_{2}^1 \le \|Q_{t_0/N^{d-d_1}} g - Q_{t/N^{d-d_1}} g \|_{L^2(\mathbb{K})}. \end{align*} $$

For $I_{3}^1$ , we will use Theorem 7.23. We rescale $I_{3}^1$ , moving from $g, R_j$ back to $f, P_j$ and write

$$ \begin{align*}I_{3}^1 = \frac{1}{N^d} \iint_{\mathbb{K}^2} f(x) \prod_{j=1}^{m-2} f(x + P_j(y))\ [\mathrm{Id} - Q_{t_0 N^{d_1}}] f (x + P_{m-1}(y)) Q_{t N^d} f(x) d\mu_{[t_1N]}(y) d\mu(x), \end{align*} $$

where the function $h'(x) = [\mathrm {Id} - Q_{t_0 N^{d_1}}] f(x)$ has the property that ${\widehat {h'}}(\xi ) = 0$ whenever $|\xi | \le (t_0 N^{d_1})^{-1}$ . Hence, for ${{\mathcal P}'} = \{P_1, \ldots , P_{m-1}\}$ ,

$$ \begin{align*} I_{3}^1 \le N^{-d} \|A^{{\mathcal P}'}_{ t_1 N}(fQ_{t N^d} f,f, \ldots, f, [\mathrm{Id} - Q_{t_0 N^{d_1}}] f) \|_{L^1(\mathbb{K})} \end{align*} $$

and so, as long as Equation (8.6) holds, Theorem 7.23 implies there exists a constant $b' = b_{{\mathcal P}'}> 0$ such that

$$ \begin{align*}\|A^{{\mathcal P}'}_{t_1N}(fQ_{t N^d} f,f, \ldots, f, h')\|_{L^1(\mathbb{K})} \lesssim_{{\mathcal P}'} \bigl(t_0/t_1^d\bigr)^{b'} \prod_{j=1}^m \|f\|_{L^{p_i}(\mathbb{K})} \le \bigl(t_0/t_1^d\bigr)^{b'} N^d \end{align*} $$

since $1/p_1 + \cdots + 1/p_{m-1} = 1$ and $\|f\|_{L^{p_i}(\mathbb {K})} \le N^{d/p_i}$ for (which follows since f is 1-bounded and supported in $[N^d]$ ). Hence,

$$ \begin{align*} I_{3}^1 \lesssim_{{\mathcal P}'} \bigl(t_0/t_1^d\bigr)^{b'} \quad \text{ if } \quad ({8.6}) \quad \text{holds}. \end{align*} $$

Step 4

We iterate, decomposing $I_1^1 = I_{1}^2 + I_{2}^2 + I_{3}^2$ , followed by decomposing $I_1^2 = I_1^3 + I_2^3 + I_3^3$ and so on. For each $0\le j \le m-1$ , we have

(8.7) $$ \begin{align} &I_1^j = \iint_{\mathbb{K}^2} g(x) \Big(\prod_{i=1}^{m-j-1} g(x+ R_i(y))\Big) \, \Big(\prod_{i=0}^jQ_{t/N^{d-d_i}} g(x)\Big)\, d\mu_{[t_1]}(y) d\mu(x), \end{align} $$

(8.8) $$ \begin{align} &I_{2}^j \le \|Q_{t_0/N^{d-d_j}} g - Q_{t/N^{d-d_j}} g \|_{L^2(\mathbb{K})} \quad \text{and} \quad I_{3}^j \lesssim_{\mathcal P} \bigl(t_0/t_1^d\bigr)^{b} \quad \text{for some} \quad b = b_{\mathcal P}> 0, \end{align} $$

again if Equation (8.6) holds. Strictly speaking, the estimate (8.8) for $I_3^j$ does not follow from Theorem 7.23 when $j=m-1$ since the proof of Theorem 7.23 assumed that the collection ${\mathcal P}$ of polynomials consisted of at least two polynomials. Nevertheless, the bound (8.8) holds when $j=m-1$ . To see this, we apply the Cauchy–Schwarz inequality and Plancherel’s theorem to see that

$$ \begin{align*} | I_3^{m-1} |^2 \ \le \ \frac{1}{N^{d}} \int_{\mathbb{K}} \bigl| \int_{\mathbb{K}} [\mathrm{Id} - Q_{t_0N^{d_{m-1}}}] f (x + P_1(y)) \, d\mu_{[t_1N ]}(y)\bigr|^2\, d\mu(x) \qquad\qquad\qquad\\= \frac{1}{N^{d}} \int_{|\xi|\ge (N^{d_{m-1}} t_0)^{-1}} |{\widehat{f}}(\xi)|^2 |m_{N,t_1}(\xi)|^2\, d\mu(\xi), \quad \text{ where } \quad m_{N, t_1}(\xi) := \int_{B_1(0)} \mathrm{e}(P_1(t_1N y)\xi)\, d\mu(y). \end{align*} $$

The oscillatory integral bound (3.1) implies that $|m_{N,t_1}(\xi )| \lesssim _{\mathcal P} (t_0/t_1)^b$ whenever $|\xi | \ge (N^{d_{m-1}} t_0)^{-1}$ , and so Equation (8.8) for $I_3^j$ follows when $j=m-1$ since $\|f\|_{L^2(\mathbb {K})}^2 \le N^d$ .

Step 5

From Equation (8.4) and the iterated decomposition of $I_1$ , we see that $t_1^{-1} I \ge A + B + C$ , where

$$ \begin{align*}A = \int_{\mathbb{K}} g(x) \prod_{j=0}^{m-1}Q_{t/N^{d-d_j}} g(x) \, d\mu(x) \ \ge \ \varepsilon^{m+1} \end{align*} $$

by Equation (8.2), and for some $C_{\mathcal P}>0$ , we have

$$\begin{align*}|B| \le C_{\mathcal P} \sum_{j=0}^{m-1} \|Q_{t_0/N^{d-d_j}} g - Q_{t/N^{d-d_j}} g \|_{L^2(\mathbb{K})} \quad \text{and} \quad |C| \ \le C_{\mathcal P} \ \bigl(t_0/t_1^d\bigr)^{b} \le \varepsilon^{m+1}/4 \end{align*}$$

if $t_0 \le c_0 \, \varepsilon ^{(m+1)/b} \, t_1^d$ and $c_0^b C_{\mathcal P}<1/4$ and Equation (8.6) holds.

Finally, we claim that we can find a triple $t_0 \ll t_1 \ll t$ of small scales such that $|B| \le \varepsilon ^{m+1}/4$ . If we are able to do this, then $I \ge \varepsilon ^{m+1} t_1/2$ and the proof is complete.

Define $v:=-C_0\log _q( c_0 \varepsilon ^{(m+1)/b})$ for some large constant $C_0\gg d$ . Choose a sequence of small scales $t_0 = q^{-\ell _j}$ and $t_1 = q^{-k_j}$ and $t=q^{-u_j}$ satisfying

(8.9) $$ \begin{align} \begin{aligned} 0\le u_1< dk_1+v < \ell_1 <u_2< dk_2+v < \ell_2 < \ldots < u_n<dk_n+v<\ell_n<\ldots \\ \text{ and } \qquad\ell_{n+1}\le \ell_n-C_0\log_q( c_0 \varepsilon^{(m+1)/b}). \end{aligned} \end{align} $$

Taking $L\in \mathbb N$ such that $L=\lfloor 16C_{\mathcal P}m^2\varepsilon ^{-2(m+1)}\rfloor +1$ we claim that there exists such that

(8.10) $$ \begin{align} C_{\mathcal P}\sum_{n=0}^{m-1} \|Q_{q^{-\ell_j} N^{-(d-d_n)}} g - Q_{q^{-u_j }N^{-(d-d_n)}} g \|_{L^2(\mathbb{K})} <\varepsilon^{m+1}/4. \end{align} $$

Indeed, suppose for a contradiction that Equation (8.10) does not hold. Then for all by the Cauchy–Schwarz inequality, we have

$$ \begin{align*}\varepsilon^{2(m+1)} \le 16C_{\mathcal P}^2m \sum_{n=0}^{m-1} \|Q_{q^{-\ell_j} N^{-(d-d_n)}} g - Q_{q^{-u_j }N^{-(d-d_n)}} g \|_{L^2(\mathbb{K})}^2. \end{align*} $$

Then

and this implies $L \le 16C_{\mathcal P}^2m^2\varepsilon ^{-2(m+1)}$ since $\|g\|_{L^2(\mathbb {K})} \le 1$ , which is impossible by our choice of L.

Therefore, there exists and a corresponding triple of scales $t_0 = q^{-\ell _j}\ll t_1 = q^{-k_j}\ll t=q^{-u_j}$ satisfying the desired properties for which Equation (8.10) is true. In particular, $|B| \le \varepsilon ^{m+1}/4$ holds.

Step 6

Furthermore, with these scales by Equation (8.9), we have $t_0= q^{-\ell _j} \gtrsim (c_0 \varepsilon ^{m+1})^{O_{\mathcal P}(m^2 \varepsilon ^{-2(m+1)})}$ . In order to ensure that Equation (8.6) holds for every iteration in the decomposition, we set

$$ \begin{align*}N(\varepsilon, {\mathcal P}) \ := \ (c_0 \varepsilon^{m+1})^{- O_{\mathcal P}( m^2 \varepsilon^{-2(m+1)})} \end{align*} $$

so that for every $N\ge N(\varepsilon , {\mathcal P})$ condition (8.6) holds. Hence,

$$ \begin{align*}I \gtrsim \varepsilon^{m+1} t_1 \gtrsim \varepsilon^{m+1} t_0 \gtrsim \varepsilon^{m+1} (c_0\varepsilon^{m+1})^{O_{\mathcal P}(m^2 \varepsilon^{-2(m+1)})}, \end{align*} $$

establishing the desired bound (8.3) with $\delta = \varepsilon ^{C_1 \varepsilon ^{-2m-2}}$ for some $C_1>0$ depending only on ${\mathcal P}$ .

This completes the proof of Theorem 1.3.