Related papers: The larger sieve and polynomial congruences

We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a…

We show a new large sieve version of the Brun-Titchmarsh theorem.

Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make…

Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a conjectural supercongruence for numbers which are coefficients in one…

The purpose of this paper is to present a syatemic study of some familes of higher-order Euler numbers and polynomials. In particular, by using the basis property of higher-order Euler polynomials for the space of polynomials of degree less…

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $\alpha \in (1/2,1)$ achieve the order at least $N^{\alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials…

This short note extends a recent result (Bonifas et al, On sub-determinants and the diameter of polyhedra, Discrete Computational Geometry, 52, 2014) of an upper bound of the diameter of a convex polytope defined by an integer matrix to a…

We give a new bound for the large sieve inequality with power moduli q^k that is uniform in k. The proof uses a new theorem due to T. Wooley from his work on efficient congruencing.

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the…

In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.

We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials…

We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our…

One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the…

This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth…