Congruences with intervals and arbitrary sets

Abstract

Given a prime p, an integer $H\in [1,p)$, and an arbitrary set ${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$, where ${\mathbb {F}} _p$ is the finite field with p elements, let $J(H,{\mathcal {M}} )$ denote the number of solutions to the congruence

$$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$

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 et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).