We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.

We discuss 1-factorizations of complete graphs that “match” a given Hadamard matrix. We prove the existence of these factorizations for two families of Hadamard matrices: Walsh matrices and certain Paley matrices.

Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.

We examine correlations of the Möbius function over $\mathbb{F}_{q}[t]$ with linear or quadratic phases, that is, averages of the form 1

$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f<n}\unicode[STIX]{x1D707}(f)\unicode[STIX]{x1D712}(Q(f))\end{eqnarray}$$

$\unicode[STIX]{x1D712}$

$\mathbb{F}_{q}$

$Q\in \mathbb{F}_{q}[x_{0},\ldots ,x_{n-1}]$

$x_{0},\ldots ,x_{n-1}$

$f=\sum _{i<n}x_{i}t^{i}$

$\unicode[STIX]{x1D707}$

$O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$

$\unicode[STIX]{x1D716}>0$

$Q$

$O(q^{-n^{c}})$

$c>0$

$Q$

$O(q^{-c^{\prime }n})$

$c^{\prime }>0$

$Q(f)$

$f^{2}$

Let $X:=\mathbb{A}_{R}^{n}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}_{K}^{n}$ under the action of an affine finite-type group scheme can be extended to a torsor over $\mathbb{A}_{R}^{n}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}_{K}^{n}$. The proof is effective. Other cases, including $X=\unicode[STIX]{x1D6FC}_{p,R}$, are also discussed.

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_{p}$. They are unified by the theme of “breaking the $3/2$ threshold”, epitomizing the previous state of the art. This concerns two questions pivotal for the sum-product theory, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of $\mathbb{F}$, as well as the number of values of the symplectic form determined by a finite subset of $\mathbb{F}^{2}$. We establish the estimate $|R[A]|\gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios, defined by triples of elements of a set $A\subset \mathbb{F}$ (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates), pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set $P\subset \mathbb{F}^{2}$, the number of distinct values of the symplectic form $\unicode[STIX]{x1D714}$ on pairs of distinct points $u,u^{\prime }$ of $P$ is $|\unicode[STIX]{x1D714}(P)|\gtrsim |P|^{2/3+c}$, with an explicit $c$. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein. Our proofs make use of specially suited applications of new incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. The technical thrust of the paper uses additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any $\mathbb{F}$, by showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^{2}$, the quantity often arising in applications of geometric incidence estimates.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.

Motivated by the Erdős–Szekeres convex polytope conjecture in $\mathbb{R}^{d}$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $n>k\geqslant 5$, what is the minimum integer $g_{k}(n)$ such that any$k$-uniform hypergraph on $g_{k}(n)$ vertices with the property that any set of $k+1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_{k}(n)>2^{cn^{k-4}}$, where $c=c(k)$.

We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$. While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

For each $n\geqslant 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^{n}$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^{n}$ times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.