arXiv subjects

Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 91 records · Page 5Linked to original sources

arXiv2020

Dynamical irreducibility of polynomials modulo primes

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes $p$ such that all iterations of $f$ are irreducible modulo $p$ is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval $[1, Q]$ as $Q \to \infty$. For this class of polynomials this gives a more precise version of a recent result of A. Ferraguti (2018), which applies to arbitrary polynomials but requires a certain assumption about their Galois group. Furthermore, under the Generalised Riemann Hypothesis we obtain a stronger bound on this density.

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arXiv2020

On the distribution of modular square roots of primes

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.

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arXiv2020

On the Dynamical System Generated by the Möbius transformation at Prime Times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map $x \mapsto (ax + b)/(cx+d)$ over a finite field of $p$ elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

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arXiv2020

Bilinear forms in Weyl sums for modular square roots and applications

Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salié sums). Our estimate has a power saving in the so-called P{ó}lya-Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in $\mathbb{F}_q$. This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted $L$-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdos-Odlyzko-Sarkozy.

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arXiv2020

On elements of large order of elliptic curves and multiplicative dependent images of rational functions over finite fields

Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo $p$ of a finite set of rational functions have large multiplicative orders

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arXiv2020

Exponential sums with sparse polynomials over finite fields

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_νx^{n_ν}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_ν$ and their differences. In particular, two new bounds for binomials are obtained, improving previous results in broad ranges of parameters.

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arXiv2020

Möbius Randomness Law for Frobenius Traces

Recently E. Bombieri and N. M. Katz (2010) have demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.

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arXiv2020

Polynomial Equations in Subgroups and Applications

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 improvement of recent results of J. Bourgain, A. Gamburd and P. Sarnak (2016) and S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski and I. V. Vyugin (2019) on the structure of solutions to the reduction of the Markoff equation $x^2 + y^2 + z^2 = 3 x yz$ modulo a prime $p$.

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arXiv2020

On Large Values of Weyl Sums

A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2πi \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients $(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of $[0,1]^d$. We also use a different technique to give similar results for sums with just one monomial $xn^d$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.

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arXiv2020

New estimates for exponential sums over multiplicative subgroups and intervals in prime fields

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 πi z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.

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arXiv2020

Bounds of Trilinear and Trinomial Exponential Sums

We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups.

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arXiv2020

Hybrid bounds on two-parametric family Weyl sums along smooth curves

We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(τx+c) ω(n)+xn$, $n=1, 2, \ldots$, with fixed $ω(T) \in \mathbb{Z}[T]$ and $τ\in \mathbb{R}$, which holds for almost all $c\in [0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $xn+y ω(n)$ for all $(x,y)\in [0,1)^2$ with $f(x,y)=z$ for a set of $z \in [0,1)$ of full Lebesgue measure, provided that $f$ is some Hölder function.

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arXiv2020

Restricted mean value theorems and metric theory of restricted Weyl sums

We study an apparently new question about the behaviour of Weyl sums on a subset $\mathcal{X}\subseteq [0,1)^d$ with a natural measure $μ$ on $\mathcal{X}$. For certain measure spaces $(\mathcal{X}, μ)$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for $μ$-almost all points of $\mathcal{X}$ the Weyl sums satisfy the square root cancellation law. Moreover we characterise the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $(\mathcal{X}, μ)$. We obtain general results, which we refine for some special spaces $\mathcal{X}$ such as spheres, moment curves and line segments.

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arXiv2020

Effective bounds on multiplicatively dependent orbits of integer polynomials modulo S-integers

We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring of integers of $\mathbb{K}$. Our results provide an effective variant of a recent result of A.Bérczes, A.Ostafe, I.E.Shparlinski and J.H.Silverman (arXiv:1811.04971) on multiplicative dependence modulo a finitely generated subgroup by eliminating the use of non-effective results by K.F.Roth and G.Faltings.

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arXiv2019

On the number of products which form perfect powers and discriminants of multiquadratic extensions

We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of $\mathbb{Q}$ and improve and generalize a result of N. Rome (2017).

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arXiv2019

On a Hybrid Version of the Vinogradov Mean Value Theorem

Given a family $φ= (φ_1, \ldots, φ_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $ρ$, we consider the mean value $$ M_{k, ρ} (φ, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ(\mathbf{x}, \mathbf{y}; N) \right|^ρd\mathbf{x} $$ of exponential sums $$ S_φ( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 πi \left(\sum_{j=1}^k x_j φ_j(n)+ \sum_{j=1}^{d-k}y_jφ_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of polynomials $φ_i(T) = T^i$, $i =1, \ldots, d$ and $k=d$ corresponds to the classical Vinaogradov mean value theorem. Here motivated by recent works of Wooley (2015) and the authors (2019) on bounds on $\sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ( \mathbf{x}, \mathbf{y}; N) \right|$ for almost all $\mathbf{x} \in [0,1]^k$, we obtain nontrivial bounds on $M_{k, ρ} (φ, N)$.

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arXiv2019

New bounds of Weyl sums

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a ``self-improving'' approach, which leads an infinite series of monotonically decreasing bound, converging to our final result.

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arXiv2019

Unlikely intersections over finite fields: polynomial orbits in small subgroups

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.

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