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Igor Shparlinski

Publications and source records attributed to Igor Shparlinski.

At least 19 recordsLinked to original sources
arXiv2026

Large sieve inequality for sums of Legendre symbols over short intervals

Using the Burgess bound and the Selberg sieve, we obtain an upper bound for the second moment of sums of Legendre symbols over intervals , with the modulus ranging over primes . The bound is nontrivial and yields a power saving in , uniformly for , provided that , where as . This may be viewed as a short-interval analogue of a result of D. R. Heath-Brown (1995) on moments of quadratic character sums over the initial interval . In particular, it implies that, for any prescribed interval of this length, the quadratic residues and non-residues are asymptotically equidistributed for almost all primes . We also establish estimates for higher moments conditionally on the Generalised Riemann Hypothesis. These bounds rely on a sharp uniform estimate for the number of tuples of integers in a shifted interval whose product is a square.

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arXiv2026

Large values of $L(\sigma,\chi)$ for subgroups of characters

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots.

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arXiv2024

Exponential Sums with Sparse Polynomials and Distribution of the Power Generator

We obtain new bounds on complete rational exponential sums with sparse polynomials modulo a prime, under some mild conditions on the degrees of the monomials of such polynomials. These bounds, when they apply, give explicit versions of a result of J. Bourgain (2005). In turn, as an application, we also obtain an explicit version of a result of J. Bourgain (2010) on national exponential sums with sparse polynomials modulo an arbitrary composite number. We then use one of these bounds to study the multidimensional distribution of the classical power generator of pseudorandom numbers, which has not been possible within previously known results.

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arXiv2024

Pointwise and correlation bounds on Dedekind sums over small subgroups

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(λ,p)$ modulo a prime $p$ with $λ$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of $L(1,χ)$ over subgroups of characters to some correlations of Dedekind sums and use our recent results to study these correlations.

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arXiv2023

Bohr sets generated by polynomials and Coppersmith's method in many variables

We obtain bounds on the average size of Bohr sets with coefficients parametrised by polynomials over finite fields and obtain a series of general results and also some sharper results for specific sets which are important for applications to computer science. In particular, we use our estimates to show that a heuristic assumption used in the many variable version of Coppersmith's method holds with high probability. We demonstrate the use of our results on the approximate greatest common divisor problem and obtain a fully rigorous version of the heuristic algorithm of H. Cohn and N. Heninger (2013).

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arXiv2023

On some matrix counting problems

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound of V. Blomer and J. Li (2022).

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arXiv2023

On a family of sparse exponential sums

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain (2005) gave a nontrivial estimate for these and more general sums. In this work we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.

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arXiv2023

Square-free values of random polynomials

The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.

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arXiv2022

An effective local-global principle for algebraic varieties and the sum product problem in finite fields

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products and reciprocals of linear forms. This allows us to make some progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev and I. D. Shkredov (2019) on an extreme case of the Erdős-Szemerédi conjecture in finite fields.

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arXiv2021

On the Skolem problem and some related questions for parametric families of linear recurrence sequences

We show that in a parametric family of linear recurrence sequences $a_1(α) f_1(α)^n + \ldots + a_k(α) f_k(α)^n$ with the coefficients $a_i$ and characteristic roots $f_i$, $i=1, \ldots,k$, given by rational functions over some number field, for all but a set of $α$ of bounded height in the algebraic closure of $\mathbb Q$, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.

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arXiv2021

Order of torsion for reduction of linearly independent points for a family of Drinfeld modules

Let $q$ be a power of the prime number $p$, let $K={\mathbb F}_q(t)$, and let $r\ge 2$ be an integer. For points ${\mathbf a}, {\mathbf b}\in K$ which are $\mathbb{F}_q$-linearly independent, we show that there exist positive constants $N_0$ and $c_0$ such that for each integer $\ell\ge N_0$ and for each generator $τ$ of ${\mathbb F}_{q^\ell}/{\mathbb F}_q$, we have that for all except $N_0$ values $λ\in{\overline{\mathbb{F}_q}}$, the corresponding specializations ${\mathbf a}, {\mathbf b}(τ)$ and ${\mathbf b}(τ)$ cannot have orders of degrees less than $c_0\log\log\ell$ as torsion points for the Drinfeld module $Φ^{(τ,λ)}:\mathbb{F}_q[T] {\longrightarrow} {\mathrm{End}}_{\overline{\mathbb{F}_q}}({\mathbb G}_a)$ (where ${\mathbb G}_a$ is the additive group scheme), given by $Φ^{(τ,λ)}_T(x)=τx+λx^q + x^{q^r}$.

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arXiv2020

Metric theory of lower bounds on Weyl sums

We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$, such that $$ \left|\sum_{n=1}^N \exp\left(2 πi\left(x_1n+\ldots+x_d n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d \ge 3$ and at least $3/2$ for $d=2$, where $c$ is a constant depending only on $d$. This improves the previous lower bound of the first and third authors for $d\ge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $xn^d$.

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arXiv2020

Metric theory of Weyl sums

We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 πi \left (x_1n+\ldots+x_d n^d\right)\right) \right|\le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full Lebesque measure. This substantially improves the previous results where similar sets have been measured in terms of the Hausdorff dimension. We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d\neq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.

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arXiv2020

Noisy polynomial interpolation modulo prime powers

We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. Similar results are known for residues modulo a large prime $p$, however the case of prime power modulus $p^k$, with small $p$ and large $k$, is new and requires different techniques. We give a deterministic polynomial time algorithm, which for almost given more than a half bits of $f(t)$ for sufficiently many randomly chosen points $t \in \mathbb Z_{p^k}^*$, recovers $f(X)$.

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arXiv2020

On a conjecture of Soundararajan

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,χ)$ for characters $χ$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,χ)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.

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arXiv2019

Congruences with intervals and arbitrary sets

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski, Michel and Sawin (2018).

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arXiv2019

Kloosterman paths of prime powers moduli, II

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/p^(n/2) converge in law in the Banach space of complex-valued continuous function on [0,1] to an explicit random Fourier series as (a,b) varies over (Z/p^nZ)^\times\times(Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case. The purpose of this work is to prove a convergence law in this Banach space as only a varies over (Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=31 is a fixed integer.

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