László Mérai

Given a matrix $A\in \mathrm {GL}_d(\mathbb {Z})$ . We study the pseudorandomness of vectors $\mathbf {u}_n$ generated by a linear recurrence relation of the form $\begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*}$ modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geqslant 1$ . We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of $\mathbf {u}_n$ and control the $p$ -adic orders of their coefficients in polynomial representations.

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).

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.

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 focus on the particular case $q=2$, which is more intricate. The bounds depend on certain natural restrictions. We also provide families of examples for which the conditions of our bounds are fulfilled. In particular, we completely classify all monomials as argument of the additive character for which our bound is applicable. Moreover, we also show that it is applicable for a large family of rational functions, which includes all reciprocal monomials.

Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some $A\in\mathcal{A}$ and $B\in \mathcal{B}$.

A function $F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n}$ , $n=2m$ , can have at most $2^{n}-2^{m}$ bent component functions. Trivial examples are vectorial bent functions from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{m}$ , seen as functions on $\mathbb {F}_{2}^{n}$ . The first nontrivial example is given in univariate form as $x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m$ (Pott et al. 2018), a few more examples of similar shape are given by Mesnager et al. 2019, and finally it has been shown that the quadratic function $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ , has $2^{n}-2^{m}$ bent components if and only if $\Lambda $ is a linearized permutation polynomial of $\mathbb {F}_{2^{m}}[x]$ (Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with $2^{n}-2^{m}$ bent components is shown, which is attained by the example $x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ . We then analyse in detail nonlinearity and differential spectrum of the class of functions $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ , which, as will be seen, requires the study of the functions $x^{2^{r}}\Lambda (x)$ . In the last part we demonstrate that this class belongs to a larger class of functions with $2^{n}-2^{m}$ Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.

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 $M_p$, $p \leq X$, 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 $(\log X)^{3/2+o(1)}$ rightmost digits of $M_p$, $p \leq X$. Previous results imply this only for about $(\log X)^{1+o(1)}$ rightmost digits.

For positive integers $K$ and $L$, we introduce and study the notion of $K$-multiplicative dependence over the algebraic closure $\overline{\mathbb{F}}_p$ of a finite prime field $\mathbb{F}_p$, as well as $L$-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions $\varphi_1,\ldots,\varphi_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $\alpha\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $\varphi_1(\alpha),\ldots,\varphi_m(\alpha)$ are $K$-multiplicatively dependent or the points $(\varrho_1(\alpha),\cdot), \ldots,(\varrho_n(\alpha),\cdot)$ are $L$-linearly dependent on the reduction of $E$ modulo $p$. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety $\mathbb{G}_{\mathrm{m}}^m \times E^n$ with the algebraic subgroups of codimension at least $2$. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases.