Related papers: On certain bilinear sums with modular square roots…
We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.
We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…
In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in $\mathbb{Z}/m\mathbb{Z}$ for arbitrary integers $m$. These results have been motivated by a wide variety of…
In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the…
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.
We establish a general large sieve inequality with sparse sets $\mathcal{S}$ of moduli in the Gaussian integers which are in a sense well-distributed in arithmetic progressions. This extends earlier work of S. Baier on the large sieve with…
The sum of square roots is as follows: Given $x_1,\dots,x_n \in \mathbb{Z}$ and $a_1,\dots,a_n \in \mathbb{N}$ decide whether $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$. It is a prominent open problem (Problem 33 of the Open Problems Project),…
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
Given a monic polynomial $f(X)\in \mathbb{Z}_m[X]$ over a residue ring $\mathbb{Z}_m$ modulo an integer $m\ge 2$ and a discrete interval $\mathcal{I} = \{1, \ldots, H\}$ of $H \le m$ consecutive integers, considered as elements of…
In this paper we obtain some refinements of a well-known result of Enestr\"o-Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve upon some results due to Aziz and Mohammad, Govil…
We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular…
We prove a lower bound for the large sieve with square moduli.
The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information…
In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.
Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these…
We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
In our previous paper [GSV2020], we proved that the complementary components of a ring domain in $\mathbb{R}^n$ with large enough modulus may be separated by an annular ring domain and applied this result to boundary correspondence problems…
In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the…
We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalises previous results of P. Corvaja and U. Zannier (2013). We also obtain a conditional…