# Congruences with intervals and arbitrary sets

## TL;DR

This paper establishes bounds on the number of solutions to certain congruences involving intervals and arbitrary sets in finite fields, with applications to character sums and Kloosterman sums, advancing understanding in analytic number theory.

## Contribution

It provides new bounds for solutions to specific congruences in finite fields, which are optimal in many parameter ranges, and applies these bounds to improve estimates of character and Kloosterman sums.

## Key findings

- Bound on solutions to congruences involving intervals and arbitrary sets.
- Optimal bounds in a wide parameter range.
- Improved estimates for character sums and Kloosterman sums.

## Abstract

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. 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).