We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope 𝔭 parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 1–25], the authors asked whether there are Besicovitch phenomena in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. Furthermore, we prove that any Kakeya set in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{2} $ or ${ \mathbb{Z} }_{p}^{2} $ is of Minkowski dimension 2.

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of $d+ 2$ points on the $(d- 1)$-sphere ${S}^{d- 1} $ are not contained in a transitive set in some ${\mathbf{R} }^{n} $.

We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least

The Littlewood conjecture states that for all (α,β)∈ℝ2. We show that with the additional factor of log q⋅log log q, the statement is false. Indeed, our main result implies that the set of (α,β) for which is of full dimension.

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

Let Q be an infinite set of positive integers. Denote by Wτ,n(Q) (respectively, Wτ,n) the set of points in dimension n≥1 that are simultaneously τ-approximable by infinitely many rationals with denominators in Q (respectively, in ℕ*). When n≥2 and τ>1+1/(n−1) , a non-trivial lower bound for the Hausdorff dimension of the liminf set Wτ,n ∖Wτ,n (Q) is established in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension and the one-dimensional analogue of the problem are also discussed.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn-close to some ∑ ni=1ωiλi, where ωi∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log   2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.

We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χ≠χ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product $ \mathbb {G}_{\mathrm {a}}\times \mathbb {G}_{\mathrm {m}}$ whose underlying group of complex points is $\mathbb {C}\times \mathbb {C}^{*}$. It shows that if a certain sequence of non-zero polynomials in $ \mathbb {Z}[X_1,X_2]$ takes small values at a point $(\xi ,\eta )$ together with their first derivatives with respect to the invariant derivation $\partial /\partial X_1 + X_2 (\partial /\partial X_2)$, then both $\xi $ and $\eta $ are algebraic over $\mathbb {Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon’s criterion for algebraic independence and compares favorably with constructions coming from Dirichlet’s box principle.

We find the best asymptotic lower bounds for the coefficient of the leading term of the ${L}_{1} $ norm of the two-dimensional axis-parallel discrepancy that can be obtained by Roth’s orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, and complex and harmonic analysis.

We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov’s pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

In this paper we generalize and extend work by Languasco, Perreli and Zaccagnini on the Montgomery–Hooley theorem in short intervals. We are thus able to obtain asymptotic formulae for the mean-square error in the distribution of general sequences in arithmetic progressions in short intervals. As applications we consider lower and upper bound approximations to the characteristic function of the primes in a short interval and an average over short intervals for the von Mangoldt function.

We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright’s result that seems to be simpler and relies upon basic estimates for exponential sums mod $p$as well as a type of resolution of singularities with good reduction in the sense of Denef.

Given a probability density, we estimate the rate of decay of the measure of the level sets of its evolutes by the Ornstein–Uhlenbeck semigroup. The rate is faster than what follows from the preservation of mass and Markov’s inequality.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.