Igor E. Shparlinski

UNSW Sydney | UNSW · Department of Pure Mathematics

Let $N_a(x)$ denote the number of primes up to $x$ for which the integer $a$ is a primitive root. We show that $N_a(x)$ satisfies the asymptotic predicted by Artin's conjecture for almost all $1\le a\le \exp((\log \log x)^2)$. This improves on a result of Stephens (1969). A key ingredient in the proof is a new short character sum estimate over the...

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. B...

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums.

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers.

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm Sp}(2g,\mathbb Z)$, which we take to be ergodic.Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this se...

We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of Kloosterman and Gauss sums due to A. Ostafe, I. E. Shparlinski and J. F. Voloch (2021).

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|= \lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\varepsilon}\rfloor$, any integer $\lambda$ can be represe...

We consider the set of m × n {m\times n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fract...

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 gave a nontrivial estimate for...

We obtain new estimates –both upper and lower bounds– on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures of C. Demeter and B. Langowski (2022), and improve some of their results.

We obtain a new bound on certain multiple sums with multidimensional Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums in very small families.

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $O(X^{n^2-1})$, where $X$ is the cardinality of $\mathcal X$. Here we show that even for arbitrary se...

Towards a well-known open question in arithmetic dynamics, L. M\'erai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials $f \in \mathbb Z[X]$, which in particular includes all quadratic polynomials, that, under some natural conditions (necessary for quadratic polynomials), the set of primes $p$, such that all iterations...

We obtain asymptotic formulas for the number of matrices in the congruence subgroup $$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$ which are of naive height at most X . Our result is uniform in a very broad range of values Q and X .

We use bounds on bilinear forms with Kloosterman fractions and improve the error term in the asymptotic formula of Balazard and Martin (Bull Sci Math 187:Art. 103305, 2023) on the average value of the smallest denominators of rational numbers in short intervals.

We obtain new bounds, pointwise and on average, for Dedekind sums s(λ,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{s}}(\lambda ,p)$$\end{document} modulo...

For an ‐bit positive integer written in binary as where , , , let us define the digital reversal of . Also let . With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of such that and are prime. We also prove that for sufficiently large , where denotes the number of prime factors counted with multiplicity...

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimat...

We estimate mixed character sums of polynomial values over elements of a finite field with sparse representations in a fixed ordered basis over the subfield . First we use a combination of the inclusion–exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large . Then we focus on the particular case ,...

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences x2≡p(modq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{doc...

In this paper, we use some of our previous results to improve an upper bound of Bayer–Fluckiger, Borello, and Jossen on the Euclidean minima of algebraic number fields. Our bound depends on the degree n of the field, its signature, discriminant, and the Hermite constant in dimension n.

Given a matrix A ∈ G L d ( Z ) A\in \mathrm {GL}_d(\mathbb {Z}) . We study the pseudorandomness of vectors u n \mathbf {u}_n generated by a linear recurrence relation of the form u n + 1 ≡ A u n ( mod p t ) , n = 0 , 1 , … , \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo p t...

We obtain upper bounds for the number of monic irreducible polynomials over $$\mathbb{Z}$$ Z of a fixed degree n and a growing height H for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower...

For a prime number p and integer x with , let denote the multiplicative inverse of x modulo p . In this paper, we are interested in the problem of distribution modulo p of the sequence and in lower bound estimates for the corresponding exponential sums. As representative examples, we state the following two consequences of the main results. For any...

We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set. \

We obtain asymptotic formulas for the number of matrices in the congruence subgroup \[ \Gamma_0(Q) = \left\{ A\in\mathrm{SL}_2(\mathbb Z):~c \equiv 0 \pmod Q\right\}, \] which are of naive height at most $X$. Our result is uniform in a very broad range of values $Q$ and $X$.

We formulate several analogs 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.

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.

In this paper, we use some of our previous results to improve an upper bound of Bayer-Fluckiger, Borello and Jossen on the Euclidean minima of algebraic number fields. Our bound depends on the degree $n$ of the field, its signature, discriminant and the Hermite constant in dimension $n$.

We use recent results about linking the number of zeros on algebraic varieties over 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. Ro...

Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $\gamma > 0$ and $x \ge 2$ we define $$H_{\gamma}(x)=\#\left\{m:~m=pq \le x, \ p<q\text{ primes }, \ \vartheta(m)\le m^{1/2+\gamma}\right\}, $$ and show that for any fixed $\eta> 0$, uniformly over $\gamma$ with $$9/20+\eta \le \gamma\le 1/2 -\eta, $$...

Let $f(X) \in {\mathbb Z}[X]$ be a polynomial of degree $d \ge 2$ without multiple roots and let ${\mathcal F}(N)$ be the set of Farey fractions of order N . We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields ${\mathbb Q}(\sqrt {f(r)})$ for $r\in {\mathcal F}(N)$...

We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.

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 $f(\mathcal I)$ d...

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.

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates...

We estimate weighted character sums with determinants ad − bc of 2 × 2 matrices modulo a prime p with entries a, b, c and d varying over the interval [1, N]. Our goal is to obtain non-trivial bounds for values of N as small as possible. In particular, we achieve this goal, with a power saving, for N ⩾ p1/8+ε with any fixed ε > 0, which is very like...

We obtain new bounds, pointwisely and on average, for Dedekind sums $s(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. As an application, we use these bounds to extend an asymptotic formula of S.~Louboutin and M.~Munsch (2022) for the second moment of Dirichlet $L$-functions over subgroups of the group of...

We show that if one selects uniformly independently and identically distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the matrices $A_1, \ldots, A_s$ are free generators for a free subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Furthermore, to show the...

Let $\varepsilon>0$ be a fixed small constant, ${\mathbb F}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb F}_p$ that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let ${\mathcal M}$ be an arbitrary subset of ${\mathbb F}_p$. If $\#{\...

For a prime number $p$ and integer $x$ with $\gcd(x,p)=1$ let $\overline{x}$ denote the multiplicative inverse of $x$ modulo $p.$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$ \overline{x}, \qquad x =1, \ldots, N, $$ and in lower bound estimates for the corresponding exponential sums. As represe...

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobeniu...

We give upper bounds on the power moments of the number of fixed points of a family of subset sum pseudorandom number generators, introduced by Rueppel (Analysis and design of stream ciphers, Springer-Verlag, Berlin, 1986).

We obtain new estimates - both upper and lower bounds - on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures of C. Demeter and B. Langowski (2022), and improve some of their results.

Given a matrix $A\in \mathrm{GL}_d(\mathbb{Z})$. We study the pseudorandomness of vectors $\mathbf{u}_n$ generated by a linear recurrent relation of the form $$ \mathbf{u}_{n+1} \equiv A \mathbf{u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, $$ modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geq 1$. We study such sequences over v...

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine $m$-tuples in finite fields, which is due to A. Dujella and M.Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).

We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.

We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n – 1) additions.

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the inclusion-exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large $q$. Then...

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.

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

We estimate weighted character sums with determinants $ad-bc $ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d $ varying over the interval $ [1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as possible. In particular, we achieve this goal, with a power saving, for $N \ge p^{1/8+\varepsilon}\ $ with any fix...

We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear forms and to some questions regarding smooth and square-free polynomials in residue classes.

We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$ with a fixed |$n\times n$| matrix |$A$| over a finite field |${{\mathbb {F}}}_q$| of |$q$| elements of multiplicative order |$\tau $|⁠. We apply o...

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic c...

We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).

We obtain a new bound in the uniform version of the Glasner property for matrices with polynomial entries, improving that of K. Bulinski and A. Fish (2021). This improvement is based on a more careful examination of complete rational exponential sums with polynomials and can perhaps be used in other questions of the similar spirit.

We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalizes previous results of P. Corvaja and U. Zannier (2013). We also obtain a conditional improvement of recent results of J. Bourgain, A. Gamburd, and P. Sarnak (2016) and S. V. Konyagin, S. V. Makarychev, I....

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error terms in our formulas depend on the Diophantine properties of the leading coefficients of these polynomials.

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by \'E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, \'E. Fouvry, E. Kowalski, Ph. Michel and D. Mili\'cevi\'c (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new es...

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal M_n(\mathbb Z; H)$, which satisfy various multiplicative relations. These problems generalise those studied in the scalar case $n=1$ by F. Pappala...

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given α∈(1/2,1) achieve the order at least Nα for infinitely many sum lengths N. For Weyl sums with polynomials of degree d⩾3 we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large va...

We prove that there exist positive constants C and c such that for any integer d⩾2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqslant 2$$\end{document} the set...

We obtain upper bounds on the cardinality of Hilbert cubes in finite fields, which avoid large product sets and reciprocals of sum sets. In particular, our results replace recent estimates of N. Hegyv\'ari and P. P. Pach (2020), which appear to be void for all admissible parameters. Our approach is different from that of N. Hegyv\'ari and P. P. Pac...

Motivated by recently developed interest to the distribution of q -ary digits of Mersenne numbers M_p = 2^p-1 , where p is prime, we estimate rational exponential sums with M_p , p \le X , modulo a large power of a fixed odd prime q . In turn this immediately implies the normality of strings of q -ary digits amongst about (\log X)^{3/2+o(1)} rightm...

We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear sums and to some questions regarding smooth and square-free polynomials in residue classes.

We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).

We obtain a new bound in the uniform version of the Glasner property for matrices with polynomial entries, improving that of K. Bulinski and A. Fish (2021). This improvement is based on a more careful examination of complete rational exponential sums with polynomials and can perhaps be used in other questions of the similar spirit.

We give a corrected version of our previous lower bound on the value set of binomials (Canad. Math. Bull., v.63, 2020, 187–196). The other results are not affected.

We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_\nu} = A^{x_{\nu+1}} + \ldots + A^{x_{2\nu}}, \quad 1 \le x_1, \ldots,x_{2\nu} \le \tau, $$ with a fixed $n\times n$ matrix $A$ over a finite field $\mathbb F_q$ of $q$ elements of multiplicative order $\tau$. We give applications of our result to obt...

We present a uniform description of sets of $m$ linear forms in $n$ variables over the field of rational numbers whose computation requires $m(n - 1)$ additions.

We obtain a nontrivial bound on the number of solutions to the equation $A^{x_1} + A^{x_2} = A^{x_3} + A^{x_4}$, $1 \le x_1,x_2,x_3,x_4 \le \tau$, with a fixed $n\times n$ matrix $A$ over a finite field ${\mathbb F}_q$ of $q$ elements of multiplicative order $\tau$. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) in their...

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 of degree $d\ge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficien...

We establish new results on equations and bilinear forms with modular square roots. The main motivation and application of these results is our new bound on the fourth moment of the error term in the asymptotic formula for the twisted second moment of half integral weight Dirichlet series on average over moduli.

For a class of polynomials f∈Z[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in {\mathbb {Z}}[X]$$\end{document}, which in particular includes all quadratic poly...

We obtain new estimates on the maximal operator applied to the Weyl sums. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).

Let q be a power of the prime number p, let K=Fq(t), and let r⩾2 be an integer. For points a,b∈K which are Fq-linearly independent, we show that there exist positive constants N0 and c0 such that for each integer ℓ⩾N0 and for each generator τ of Fqℓ/Fq, we have that for all except N0 values λ∈Fq‾, the corresponding specializations a(τ) and b(τ) can...

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

We obtain a new bound for trilinear exponential sums with Kloosterman fractions which in some ranges of parameters improves that of S. Bettin and V. Chandee (2018). We also obtain a similar result for more general sums.

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter a...