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Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 37 records · Page 2Linked to original sources
arXiv2025

On quantum ergodicity for higher dimensional cat maps

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 sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.

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arXiv2025

Multiple sums with the Möbius function

We establish nontrivial bounds for bilinear sums involving the Möbius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as Möbius-function analogues of the ternary Goldbach problem. By contrast, the binary versions of our results remain out of reach, much like the binary Goldbach problem. Nevertheless, we make partial progress in this direction by restricting the range of the third variable as far as possible.

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arXiv2025

Exponential sums over integers without large prime divisors

We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer $ν\ne 0$, we also obtain new bounds on exponential sums with $ν$-th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type~I and Type~II bounds. For $ν=1$ we use the classical bounds of Vinogradov (1937), while for $ν\neq 1$ we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).

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arXiv2025

Estimates for trilinear and quadrilinear character sums

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to an apparently new problem on oscillations of characters on differences between Farey fractions is given. Other applications include a modular analogue of a multiplicative hybrid problem of Iwaniec and Sárközy (1987) and the solvability of some prime type equations with constraints.

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arXiv2025

Mean value theorems for rational exponential sums

We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.

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arXiv2025

Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros

We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a classical idea of P. Erdős (1935). We then use similar ideas to get a tight upper bound on the number of linear recurrence sequences which attain a zero value.

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arXiv2025

Moments and non-vanishing of $L$-functions over thin subgroups

We obtain an asymptotic formula for all moments of Dirichlet $L$-functions $L(1,χ)$ modulo $p$ when averaged over a subgroup of characters $χ$ of size $(p-1)/d$ with $φ(d)=o(\log p)$. Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also use our ideas to get an asymptotic formula for the second moment of $L(1/2,χ)$ over subgroups of characters of similar size. This leads to non-vanishing results in this family where the proportion obtained depends on the height of the smallest rational number lying in the dual group. Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes $p$. Our method relies on pointwise and average estimates on small solutions of linear congruences which in turn leads us to use and modify some results for product sets of Farey fractions.

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arXiv2025

Number of characteristic polynomials of matrices with bounded height

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 distinct irreducible characteristic polynomials which correspond to these matrices and thus on the number of distinct eigenvalues of these matrices. In particular, we improve some results of A.~Abrams, Z.~Landau, J.~Pommersheim and N.~Srivastava (2022).

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arXiv2025

Counting matrices over finite rank multiplicative groups

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We obtain upper bounds, in terms of the size of $\mathcal A$, on the number of such matrices of a given rank, with a given determinant and with a prescribed characteristic polynomial. In particular, in the case of ranks, our results can be viewed as a statistical version of work by Alon and Solymosi (2003).

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arXiv2025

Multiplicative dependence in linear recurrence sequences

For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the corresponding elements $u(n_1), \ldots, u(n_s)$ in the sequence are multiplicatively dependent.

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arXiv2024

On Nonzero Coefficients of Binary Cyclotomic Polynomials

Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $γ> 0$ and $x \ge 2$ we define $$H_γ(x)=\#\left\{m:~m=pq \le x, \ p 0$, uniformly over $γ$ with $$9/20+η\le γ\le 1/2 -η, $$ we have an asymptotic formula $$ H_γ(x)\sim C(γ)x^{1/2+γ}/ \log x, \qquad x \to \infty, $$ where $C(γ)> 0$ is an explicit constant depending only on $γ$. This extends the previous result of {É}.~Fouvry (2013), which has $12/25$ instead of $9/20$.

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arXiv2024

Sums of Kloosterman sums over square-free and smooth integers

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.

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arXiv2024

On the Waring problem with Dickson polynomials modulo a prime

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).

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arXiv2024

Binary and ternary congruences involving intervals and sets modulo a prime

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 $λ$ can be represented in the form $$ \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2,3. $$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon}\rfloor$, then any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2. $$

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arXiv2024

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

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 a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic polynomial $f \in \mathbb Z[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. Using this result, among others, we obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb Z; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb Q)$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety.

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arXiv2024

On the determinants of matrices with elements from arbitrary sets

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 sets $\mathcal X\subseteq \mathbb R$,some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.

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arXiv2024

On the frequency of primes preserving dynamical irreducibility of polynomials

Towards a well-known open question in arithmetic dynamics, L. Mérai, 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 of $f$ are irreducible modulo $p$, is of relative density zero, with an explicit estimate on the rate of decay. This result relies on some bounds on character sums via the Brun sieve. Here we use the Selberg sieve and in some cases obtain a substantial quantitative improvement.

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