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DCDS-S

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.

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