Fractional parts of Dedekind sums

TL;DR

This paper proves a uniform distribution result for the fractional parts of Dedekind sums using advanced bounds on exponential sums, extending previous work and analyzing the least denominator of these sums on average.

Contribution

It introduces a new uniformity of distribution result for Dedekind sums and studies the average least denominator using novel techniques.

Findings

Established a uniform distribution of fractional parts of Dedekind sums.

Extended earlier results by Myerson and Vardi.

Analyzed the average least denominator of Dedekind sums.

Abstract

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec~(1997) on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums $s(m,n)$ with $m$ and $n$ running over rather general sets. Our result extends earlier work of Myerson (1988) and Vardi (1987). Using different techniques, we also study the least denominator of the collection of Dedekind sums $\bigl\{s(m,n):m\in(\mathbb Z/n \mathbb Z)^*\bigr\}$ on average for $n\in[1,N]$.