Linear Congruences with Ratios

TL;DR

This paper develops new bounds for exponential sums involving ratios of integers to derive an asymptotic formula for the number of solutions to certain linear congruences with variables from general sets.

Contribution

It introduces novel bounds on double exponential sums with ratios, enabling asymptotic analysis of solutions to linear congruences in broader contexts.

Findings

Derived asymptotic formula for solutions to linear congruences

Established new bounds for exponential sums with ratios

Extended analysis to variables from general sets

Abstract

We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences $$ \sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p, $$ with variables from rather general sets.