Triple sums of Kloosterman sums and the discrepancy of modular inverses

TL;DR

This paper studies the distribution of modular inverses within large intervals, providing bounds on their discrepancy measures and revealing deviations from randomness, supported by new bounds on triple sums of Kloosterman sums.

Contribution

It introduces new bounds for triple sums of Kloosterman sums and applies them to analyze the discrepancy of modular inverses, highlighting non-random distribution patterns.

Findings

Bounds for discrepancy measures of modular inverses

Deviations from uniform distribution observed

New bounds for triple sums of Kloosterman sums

Abstract

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums.