The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum

Abstract

This paper explores the proof by J. Bourgain, H. Furstenberg, Y. Katznelson and D.S. Ornstein of their return times theorem [2] and lights a corner in it regarding the role of auto-correlation. As for pointwise convergence, this was already observed in [5], and here we exploit the opportunity to write down the proof. This yields a more intrinsic characterization of the sequences satisfying the pointwise theorem. Then we proceed and obtain a characterization linked to auto-correlation also to sequences satisfying the mean theorem - by that theorem those were already known to be exactly the sequences with an empty Fourier-Bohr spectrum. Some further investigation is done and examples are provided regarding generic sequences satisfying the pointwise theorem for which the measure on the circle that the auto-correlation function represents (by Fourier transform) is not atomless, and also regarding the existence of sequences that satisfy the mean theorem but not the pointwise one.