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 $$x\equiv yr(\mathrm{m}\mathrm{o}\mathrm{d}p);x,y\in \mathbb{N},x,y\le H,r\in \text{cU},$$ for certain ranges of $H$ and $\mathrm{\mid }\text{cU}\mathrm{\mid }$, where $\text{cU}$ 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 $$x^n\equiv \lambda (\mathrm{m}\mathrm{o}\mathrm{d}p);x\in \mathbb{N},L<x<L+p\mathrm{/}n,$$ 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 $\text{C}$, then the congruence $ x^{f(x)}\equiv 1\pmod p, ,x\in $\mathbb{N}$, ,x\le p,$hasatmost$p^{\frac{1}{3}-c}$solutionsas$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 $xg^y \pmod p$withpositiveintegers$x<p^{5/8+\varepsilon}$and$y<p^{3/8}$\mathrm{.}Here$g$denotesaprimitiverootmodulo$p$. We also prove that almost all the residue classes modulo $p$ can be represented in the form $xyz{g}^t(\mathrm{m}\mathrm{o}\mathrm{d}p)$ with positive integers $x,y,z,t<p^{1\mathrm{/}4+\epsilon }$.