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

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 $x^n\equiv λ\pmod p; \quad x\in \N, \quad L<x<L+p/n,$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n, λ$ and $L$, for some absolute constant $c>0$. This implies, in particular, that if $f(x)\in \Z[x]$ is a fixed polynomial without multiple roots in $\C$, then the congruence $x^{f(x)}\equiv 1\pmod p, \,x\in \mathbb{N}, \,x\le p,$ 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 $xg^y \pmod p$ with positive integers $x<p^{5/8+\varepsilon}$ 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 $xyzg^t \pmod p$ with positive integers $x,y,z,t<p^{1/4+\varepsilon}$.

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