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

TL;DR

This paper analyzes the return times theorem and auto-correlation in sequences, providing intrinsic characterizations and examples, especially focusing on sequences with an empty Fourier-Bohr spectrum and their convergence properties.

Contribution

It offers a new intrinsic characterization of sequences satisfying the pointwise return times theorem and links auto-correlation to sequences with an empty Fourier-Bohr spectrum.

Findings

Sequences with an empty Fourier-Bohr spectrum satisfy the mean theorem.

Examples of sequences satisfying the pointwise theorem with non-atomless auto-correlation measures.

Existence of sequences satisfying the mean but not the pointwise theorem.

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.