A Limit Theorem for Birkoff sums of non-integrable functions over rotations

TL;DR

This paper establishes a limit theorem for Birkhoff sums of functions with singularities over rotations, showing the normalized sums converge in distribution as the number of terms grows large.

Contribution

It proves a joint limiting distribution for non-renormalized Birkhoff sums of singular functions over rotations, extending understanding of their asymptotic behavior.

Findings

Normalized Birkhoff sums converge in distribution as N approaches infinity.

A limiting distribution exists for certain trigonometric sums.

The results apply to functions with a 1/x singularity over rotations.

Abstract

We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(\alpha,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $\alpha\in [0,1)$ is the rotation number and $(\alpha, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(\alpha,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.