Limit laws for cotangent and Diophantine sums
Bence Borda, Lorenz Frühwirth, Manuel Hauke
TL;DR
This work establishes limit laws with convergence rates for ergodic sums of the form $\sum_{n=1}^N f(n\alpha)/n^p$ where $f$ has a power-type singularity at integers and $\alpha$ is randomly distributed with an absolutely continuous density. By combining Schmidt’s metric Diophantine method, quantitative Gauss–Kuzmin mixing, and $\psi$-mixing techniques, the authors derive explicit stable-law limits for both order-1 and higher-order singularities, including precise centering terms and rates of convergence in the Kolmogorov metric. In particular, they obtain a standard Cauchy limit for $\frac{1}{\log N}\sum_{n=1}^N \cot(\pi n\alpha)/n$, and joint $\mathrm{Stab}(1,1)$ limits for the positive/negative parts, with detailed descriptions for higher $p$ where the limits are $\mathrm{Stab}(1/p,1)$. The results cover cotangent sums related to Dedekind sums and Diophantine sums involving reciprocals of fractional parts, highlighting deep connections between Diophantine approximation, continued fractions, and probabilistic limit theorems in a dynamical setting.
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 $\sum_{n=1}^N f(n α)/n^p$ 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 $ψ$-mixing random variables.