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M. Z. Garaev

Publications and source records attributed to M. Z. Garaev.

25 records · Page 2Linked to original sources
arXiv2007

An explicit sum-product estimate in $\mathbb{F}_p$

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of $\mathbb{F}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain, Glibichuk, Konyagin on the size of $\max\{|A+A|, |AA|\}.$ In particular, our result implies that if $1<|A|\le p^{7/13}(\log p)^{-4/13},$ then $$ \max\{|A+A|, |AA|\}\gg \frac{|A|^{15/14}}{(\log|A|)^{2/7}} . $$

math.NT↗
arXiv2006

Waring problem with the Ramanujan $τ$-function

Let $τ(n)$ be the Ramanujan $τ$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}τ(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$

math.NT↗
arXiv2005

The large sieve for $2^{[O(n^{15/14+o(1)})]}$ modulo primes

Let $λ$ be a fixed integer, $λ\ge 2.$ Let $s_n$ be any strictly increasing sequence of positive integers satisfying $s_n\le n^{15/14+o(1)}.$ In this paper we give a version of the large sieve inequality for the sequence $λ^{s_n}.$ In particular, we prove that for $π(X)(1+o(1))$ primes $p, \ p\le X,$ the numbers $$ λ^{s_n},\quad n\le X(\log X)^{2+ε} $$ are uniformly distributed modulo $p.$

math.NT↗
arXiv2005

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.

math.NT↗
arXiv2005

On a symmetric congruence and its applications

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied to obtain new information on the exceptional set of the multiplication table problem in a residue ring modulo $m$ and a new bound for a double trigonometric sum with an exponential function.

math.NT↗