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

2016/06/10 by David Constantine, Constantine, David, Joanna Furno +1
Mathematics · #11K50 #37A30 #47A35 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1606.03441

openalex publication_date 2016/06/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We prove some results on the behavior of infinite sums of the form Σf∘ Tⁿ(x)(1)/(n), 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. We also show that there are Liouville α for which the sum converges everywhere.

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