Everywhere divergence of the one-sided ergodic Hilbert transform and Liouville numbers

TL;DR

This paper investigates the divergence and convergence behavior of a specific ergodic Hilbert transform variant for irrational circle rotations, revealing that Liouville numbers can lead to both outcomes depending on the function.

Contribution

It demonstrates that for certain functions, Liouville numbers can cause the ergodic Hilbert transform sum to diverge or converge everywhere, highlighting nuanced behavior in ergodic sums.

Findings

Liouville numbers can induce divergence of the sum for some functions.

Liouville numbers can also lead to convergence for other functions.

The behavior depends on the specific function and Liouville number considered.

Abstract

We prove some results on the behavior of infinite sums of the form $\Sigma f\circ T^n(x)\frac{1}{n}$, where $T:S^1\to S^1$ is an irrational circle rotation and $f$ is a mean-zero function on $S^1$. In particular, we show that for a certain class of functions $f$, there are Liouville $\alpha$ for which this sum diverges everywhere. We also show that there are Liouville $\alpha$ for which the sum converges everywhere.