The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911326914215936 |
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| author | Tal, Matan |
| author_facet | Tal, Matan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07453 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum Tal, Matan Dynamical Systems Probability 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. |
| title | The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2501.07453 |