Limit laws for cotangent and Diophantine sums

TL;DR

This paper establishes limit laws and convergence rates for sums involving functions with power singularities over circle rotations, with applications to cotangent sums and Diophantine approximation.

Contribution

It proves new limit laws with explicit convergence rates for sums with power singularities, extending previous ergodic results to more general sums related to number theory.

Findings

Established limit laws with convergence rates for sums with singularities

Applied results to cotangent sums and fractional part reciprocals

Connected ergodic averages to Diophantine approximation problems

Abstract

Limit laws for ergodic averages with a power singularity over circle rotations were first proved by Sinai and Ulcigrai, as well as Dolgopyat and Fayad. In this paper, we prove limit laws with an estimate for the rate of convergence for the sum $\sum_{n=1}^N f(n \alpha)/n^p$ in terms of a $1$-periodic function $f$ with a power singularity of order $p \ge 1$ at integers. Our results apply in particular to cotangent sums related to Dedekind sums, and to sums of reciprocals of fractional parts, which appear in multiplicative Diophantine approximation. The main tools are Schmidt's method in metric Diophantine approximation, the Gauss-Kuzmin problem and the theory of $\psi$-mixing random variables.