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

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.