New York Journal of Mathematics
Volume 23 (2017) 273-294


David Constantine and Joanna Furno

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

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Published: March 2, 2017
Keywords: Ergodic Hilbert transform, circle rotation, Liouville numbers, Birkhoff's theorem, continued fractions, discrepancy, Denjoy-Koksma Lemma
Subject: 37E10, 47A35, 44A15, 11A55, 11J70

We prove some results on the behavior of infinite sums of the form
∑ f ∘ 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 and Liouville α for which the sum converges everywhere.

Author information

David Constantine:
Department of Mathematics and Computer Science, Wesleyan University, 265 Church St., Middletown, CT 06459

Joanna Furno:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford, LD 270 Indianapolis, IN 46202