The modulated ergodic Hilbert transform

In this article we study the ergodic Hilbert transform modulated by bounded sequences. We prove that sequences satisfying some variation conditions and are universally good for ordinary ergodic averages, such as the sequences defined by the Fourier coefficients of $L_p$ functions, are universally good modulating sequences for the ergodic Hilbert transform. We also prove that sequences belonging to the subfamily $B_1^{\alpha} $ of the two-sided bounded Besicovitch class $B_1$ are good modulating sequences for the ergodic Hilbert transform.