On a symmetric congruence and its applications

TL;DR

This paper derives an asymptotic formula for solutions to a specific congruence with four variables in residue classes modulo a large integer m, and applies it to problems in multiplication tables and exponential sums.

Contribution

It introduces a new asymptotic formula for a congruence involving four variables and applies it to improve bounds in multiplication table problems and exponential sums.

Findings

New asymptotic formula for solutions to a four-variable congruence

Improved bounds for the exceptional set in multiplication table problems

Enhanced estimates for double trigonometric sums with exponential functions

Abstract

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied to obtain new information on the exceptional set of the multiplication table problem in a residue ring modulo $m$ and a new bound for a double trigonometric sum with an exponential function.