Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

Abstract

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$x^n\equiv \lambda (\mathrm{mod}p);x\in \mathbb{N},L<x<L+p/n,$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n,\lambda$ and $L$, for some absolute constant $c>0$. This implies, in particular, that if $f(x)\in \mathbb{Z}[x]$ is a fixed polynomial without multiple roots in $\C$, then the congruence $x^{f(x)}\equiv 1(\mathrm{mod}p),x\in \mathbb{N},x\le p,$ has at most $p^{\frac{1}{3}-c}$ solutions as $p\to \infty$, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo $p$ can be represented in the form $x{g}^y(\mathrm{mod}p)$ with positive integers $x<p^{5/8+\epsilon }$ and $y<p^{3/8}$. Here $g$ denotes a primitive root modulo $p$. We also prove that almost all the residue classes modulo $p$ can be represented in the form $xyz{g}^t(\mathrm{mod}p)$ with positive integers $x,y,z,t<p^{1/4+\epsilon }$.

25 pages. In the revised version we give more applications of the main result

References in corpus (1)

• Some new inequalities in additive combinatorics

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