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Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 55 records · Page 3Linked to original sources
arXiv2024

Uniform bounds for the density in Artin's conjecture on primitive roots

We consider Artin's conjecture on primitive roots over a number field $K$, reducing an algebraic number $α\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(α)$ counting the proportion of the primes of $K$ for which $α$ is a primitive root. This density ${\mathrm{dens}}(α)$ is a rational multiple of an Artin constant $A(τ)$ that depends on the largest integer $τ\geq 1$ such that $α\in (K^\times)^τ$. The aim of this paper is bounding the ratio ${\mathrm{dens}}(α)/A(τ)$, under the assumption that ${\mathrm{dens}}(α)\neq 0$. Over $\mathbb Q$, this ratio is between $2/3$ and $2$, these bounds being optimal. For a general number field $K$ we provide upper and lower bounds that only depend on $K$.

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arXiv2023

On the greatest common divisor of integer parts of polynomials

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 are of various strengths depending on the Diophantine properties of the leading coefficients of these polynomials.

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arXiv2023

Reversible primes

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j}, $$ the digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a \text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$ \left|\{a \in \mathcal{B}_n:~ \max \{Ω(a), Ω(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ where $Ω(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2πi (αa + \vartheta \overleftarrow{a})\right) \quad(α,\vartheta \in\mathbb{R}). $$

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arXiv2023

Counting elements of the congruence subgroup

We obtain asymptotic formulas for the number of matrices in the congruence subgroup \[ Γ_0(Q) = \left\{ A\in\mathrm{SL}_2(\mathbb Z):~c \equiv 0 \pmod Q\right\}, \] which are of naive height at most $X$. Our result is uniform in a very broad range of values $Q$ and $X$.

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arXiv2023

Euclidean minima of algebraic number fields

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 $n$.

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arXiv2023

Counting embeddings of free groups into $\mathrm{SL}_2(\mathbb{Z})$ and its subgroups

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 flexibility of our method we do similar counting for matrices from the congruence subgroup $Γ_0(Q)$ uniformly with respect to the positive integer $Q\le X$. This improves and generalises a result of E. Fuchs and I. Rivin (2017) which claims that the probability is $1 + o(1)$. We also disprove one of the statements in their work that has been used to deduce their claim.

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arXiv2023

On some congruences and exponential sums

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 $\#{\mathcal M} >p^{1/3+\varepsilon}$ and $H\ge p^{2/3}$ or if $\#{\mathcal M} >p^{3/5+\varepsilon}$ and $H\ge p^{3/5+\varepsilon}$ then all, but $O(p^{1-δ})$ elements of ${\mathbb F}_p$ can be represented in the form $hm$ with $h\in [1, H]$ and $m\in {\mathcal M}$, where $δ> 0$ depends only on $\varepsilon$. Furthermore, let $ {\mathcal X}$ be an arbitrary interval of length $H$ and $s$ be a fixed positive integer. If $$ H> p^{17/35+\varepsilon}, \quad \#{\mathcal M} > p^{17/35+\varepsilon}. $$ then the number $T_6(λ)$ of solutions of the congruence $$ \frac{m_1}{x_1^s}+ \frac{m_2}{x_2^s}+ \frac{m_3}{x_3^s}+\frac{m_4}{x_4^s}+ \frac{m_5}{x_5^s}+\frac{m_6}{x_6^s} \equiv λ\mod p, \qquad m_i\in {\mathcal M}, \quad \ x_i \in {\mathcal X}, \quad i =1, \ldots, 6, $$ satisfies $$ T_6(λ)=\frac{H^6(\#{\mathcal M})^6}{p}\left(1+O(p^{-δ})\right), $$ where $δ> 0$ depends only on $s$ and $\varepsilon$.

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arXiv2023

Bounds on bilinear forms with Kloosterman sums

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 for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet $L$-functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

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arXiv2023

On the distribution of modular inverses from short intervals

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 representative examples, we state the following two consequences of the main results. For any fixed $A > 1$ and for any sufficiently large integer $N$ there exists a prime number $p$ with $$ (\log p)^A \asymp N $$ such that $$ \max_{\gcd(a,p)=1}\left|\sum_{x\le N}{\mathbf e}_p(a\overline{x})\right|\gg N. $$ For any fixed positive $γ< 1$ there exists a positive constant $c$ such that the following holds: for any sufficiently large integer $N$ there is a prime number $p > N$ such that $$ N > \exp\left(c(\log p\log\log p)^{γ/(1+γ)}\right) $$ and $$ \max_{\gcd(a,p)=1} \left|\sum_{x\le N}{\mathbf e}_p(a \overline{x})\right|\gg N^{1-γ}. $$

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arXiv2023

Local mean value estimates for Weyl sums

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.

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arXiv2023

On character sums with determinants

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 fixed $\varepsilon>0$, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of $N$.

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arXiv2023

Distribution of recursive matrix pseudorandom number generator modulo prime powers

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 very short segments of length which is not accessible via previously used methods. Our technique is based on the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002). This is combined with some ideas from the work of I. E. Shparlinski (1978) which allows to construct polynomial representations of the coordinates of $\mathbf{u}_n$ and control the $p$-adic orders of their coefficients in polynomial representation.

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arXiv2023

Functional graphs of families of quadratic polynomials

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 Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.

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arXiv2023

On the distribution of multiplicatively dependent vectors

In this paper, we study the distribution of multiplicatively dependent vectors. For example, although they have zero Lebesgue measure, they are everywhere dense both in $\mathbb{R}^n$ and $\mathbb{C}^n$. We also study this property in a more detailed manner by considering the covering radius of such vectors.

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arXiv2023

On oracle factoring of integers

We present an oracle factorisation algorithm which finds a nontrivial factor of almost all positive integers $n$ based on the knowledge of the number of points on certain elliptic curves in residue rings modulo $n$.

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arXiv2023

On the number of Diophantine $m$-tuples in finite fields

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).

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