 

David Constantine and Joanna Furno
Everywhere divergence of the onesided ergodic Hilbert transform for circle rotations by Liouville numbers view print


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


Abstract
We prove some results on the behavior of infinite sums of the form
∑ f ∘ T^{n}(x)(1/n),
where T:S^{1}→ S^{1} is an irrational circle rotation and f is a meanzero function on S^{1}. 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
dconstantine@wesleyan.edu
Joanna Furno:
Department of Mathematical Sciences, Indiana UniversityPurdue University Indianapolis, 402 N. Blackford, LD 270 Indianapolis, IN 46202
jfurno@iupui.edu

