Dedekind sums with arguments near certain transcendental numbers

TL;DR

This paper investigates the asymptotic behavior of Dedekind sums evaluated at convergents of a specific transcendental number, revealing infinitely many transcendental cluster points within certain intervals.

Contribution

It introduces a novel analysis of Dedekind sums at convergents of a particular transcendental number, showing the existence of infinitely many transcendental cluster points.

Findings

Infinitely many transcendental cluster points of Dedekind sums.

Existence of constant-length intervals with infinitely many such points.

Asymptotic behavior of Dedekind sums near transcendental numbers.

Abstract

We study the asymptotic behaviour of the classical Dedekind sums $s(s_k/t_k)$ for the sequence of convergents $s_k/t_k$ $k\ge 0$, of the transcendental number \BD \sum_{j=0}^\infty\frac {1}{b^{2^j}},\ b\ge 3. \ED In particular, we show that there are infinitely many open intervals of constant length such that the sequence $s(s_k/t_k)$ has infinitely many transcendental cluster points in each interval.