Jörg Schmeling: Everywhere divergence of the one-sided ergodic Hilbert transform

For a given number alpha in (0; 1) and a 1-periodic function f, we study the convergence of the series \(\sum n=1^\infty f(x+n )/n\), called one-sided Hilbert transform relative to the rotation x -> x+alpha mod 1. Among others, we prove that for any non-polynomial function of class C^2 having Taylor- Fourier series (i.e. Fourier coefficients vanish on Z_-), there exists an irrational number alpha (actually a residual set of alphas) such that the series diverges for all x. We also prove that for any irrational number alpha , there exists a continuous function f such that the series diverges for all x.
The convergence of general series \(\sum n=1^\infty a_nf(x+n)\) is also discussed in different cases involving the diophantine property of the number alpha and the regularity of the function f.