Everywhere divergence of the one-sided ergodic Hilbert transform for circle rotations by Liouville numbers

摘要

We prove some results on the behavior of infinite sums of the form ∑ f ° Tn(x)(1), where T:S1→ S1 is an irrational circle rotation and f is a mean-zero function on S1. In particular, we show that for a certain class of functions f, there are Liouville α for which this sum diverges everywhere and Liouville α for which the sum converges everywhere.