Multiplicative Congruences with Variables from Short Intervals

TL;DR

This paper improves bounds on the number of solutions to certain multiplicative congruences with variables from short intervals and demonstrates the existence of elements with large order in short intervals for almost all primes.

Contribution

It provides stronger bounds for solutions to multiplicative congruences and shows that for almost all primes, large order elements can be found in short intervals.

Findings

Stronger bounds on solutions to multiplicative congruences.

Existence of large order elements in short intervals for almost all primes.

Results hold for almost all primes and fixed parameters.

Abstract

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval.