Congruences with intervals and arbitrary sets

Abstract

Given a prime $p$, an integer $H\in [1,p)$, and an arbitrary set $\mathcal{M}\subseteq {\mathbb{F}}_p^{\ast }$, where ${\mathbb{F}}_p$ is the finite field with $p$ elements, let $J(H,\mathcal{M})$ denote the number of solutions to the congruence $$xm\equiv yn\mathrm{mod}p$$ for which $x,y\in [1,H]$ and $m,n\in \mathcal{M}$. In this paper, we bound $J(H,\mathcal{M})$ in terms of $p$, $H$ and the cardinality of $\mathcal{M}$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski, Michel and Sawin (2018).