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

TL;DR

This work analyzes how sequences interact with stationary processes through two cancellation notions: pointwise cancellation and mean cancellation. It provides intrinsic characterizations via auto-correlation and links mean cancellation to having an empty Fourier-Bohr spectrum, while also extending these ideas to a mean-cancellation framework with unit-circle spectral convergence. The main results establish equivalences between spectral/auto-correlation conditions and cancellation properties, and several constructions illustrate nuances: sequences with mean cancellation but not pointwise cancellation, and generic points whose fibers fail to be generic in any joint system. The paper also uses Hochman’s tall-covers technique to demonstrate the subtle separation between genericity and cancellation phenomena in weak-mixing contexts, enriching our understanding of return times and correlation structures in ergodic theory.

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.