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I. E. Shparlinski

Publications and source records attributed to I. E. Shparlinski.

15 recordsLinked to original sources
arXiv2023

Additive energy of polynomial images

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 $\mathbb{Z}_m$, we obtain a new upper bound for the additive energy of the set $f(\mathcal I)$, where $f(\mathcal I)$ denotes the image set $f(\mathcal I) = \{f(u):~u \in \mathcal I\}$. We give an application of our bounds to multiplicative character sums, improving some previous result of Shkredov and Shparlinski~(2018).

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arXiv2022

Chowla and Sarnak Conjectures for Kloosterman Sums

We formulate several analogues of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.

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arXiv2020

Counting solvable $S$-unit equations

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

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arXiv2018

Counting integers with a smooth totient

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $φ(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.

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arXiv2018

Sums of algebraic trace functions twisted by arithmetic functions

We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function required in recent results of {É}.~Fouvry, E.~Kowalski and P.~Michel (2014) and E.~Kowalski, P. ~Michel and W.~Sawin (2018), respectively.

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arXiv2016

Cancellations Amongst Kloosterman Sums

We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel, and Milićević (2014) and Fouvry, Kowalski and Michel (2014).

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arXiv2016

Prescribing the binary digits of squarefree numbers and quadratic residues

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and Sárközy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics.

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arXiv2008

On Quadratic Fields Generated by Discriminants of Irreducible Trinomials

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $Δ_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n \equiv 1 \pmod 4$, the quadratic fields $\Q(\sqrt{Δ_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\Q$ and $|Δ_n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.

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arXiv2008

On a Generalised Lehmer Problem for Arbitrary Powers

We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang.

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arXiv2007

Density of non-residues in Burgess-type intervals and applications

We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.

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arXiv2007

Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average

We prove that the set of Farey fractions of order $T$, that is, the set $\{α/β\in \Q : \gcd(α, β) = 1, 1 \le α, β\le T\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \ge p^{1/2 +\eps}$ for any fixed $\eps>0$. We apply this to obtain upper bounds for the Lang--Trotter conjectures on Frobenius traces and Frobenius fields ``on average'' over a one-parametric family of elliptic curves.

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arXiv2007

Visible Points on Curves over Finite Fields

For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$, with the additional condition $\gcd(x,y)=1$. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over $a$ for a fixed prime $p$, and also on average over $p$ for a fixed integer $a$.

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arXiv2006

Distribution of modular inverses and multiples of small integers and the Sato--Tate conjecture on average

We show that, for sufficiently large integers $m$ and $X$, for almost all $a =1, ..., m$ the ratios $a/x$ and the products $ax$, where $|x|\le X$, are very uniformly distributed in the residue ring modulo $m$. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over $r$ and $s$, ranging over relatively short intervals, the distribution of Kloosterman sums $$ K_{r,s}(p) = \sum_{x=1}^{p-1} \exp(2 πi (rn + sn^{-1})/p), $$ for primes $p\le T$ is in accordance with the Sato--Tate conjecture.

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arXiv2006

On the distribution of Kloosterman sums

For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \sum_{x\in \F_p^*} \exp(2 πi (x + ax^{-1})/p), \qquad a \in \F_p^*, $$ over a finite field of $p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums $K_{p}(a)$ when $a$ runs through $\F_p^*$ is in accordance with the Sato--Tate conjecture. Here we show that the same holds where $a$ runs through the sums $a = u+v$ for $u \in \cU$, $v \in \cV$ for any two sufficiently large sets $\cU, \cV \subseteq \F_p^*$. We also improve a recent bound on the nonlinearity of a Boolean function associated with the sequence of signs of Kloosterman sums.

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