On the convergence of the rotated one-sided ergodic Hilbert transform

摘要

Sufficient conditions have been given for the convergence in norm and a. e. of the ergodic Hilbert transform (Gaposhkin in Theory Probab Appl 41:247-264, 1996; Cohen and Lin in Characteristic functions, scattering functions and transfer functions, pp 77-98, Birkh?user, Basel, 2009; Cuny in Ergod Theory Dyn Syst 29:1781-1788, 2009). Here we apply these conditions to the rotated ergodic Hilbert transform, where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ's for which this series does not converge is at most 2 and give examples where this bound is attained.