Correlations of the Thue--Morse sequence
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916585858400256 |
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| author | Baake, Michael Coons, Michael |
| author_facet | Baake, Michael Coons, Michael |
| contents | The pair correlations of the Thue--Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue--Morse sequence with general real weights are effectively determined by a single value of the balanced $2$-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue--Morse sequence vanish, and that, for any even $n$, the $n$-point correlations of the balanced Thue--Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue--Morse system. We finish by showing how the correlations of the Thue--Morse system with general real weights can be derived from the balanced $2$-point correlations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07102 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Correlations of the Thue--Morse sequence Baake, Michael Coons, Michael Dynamical Systems Number Theory 37B10, 52C23 The pair correlations of the Thue--Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue--Morse sequence with general real weights are effectively determined by a single value of the balanced $2$-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue--Morse sequence vanish, and that, for any even $n$, the $n$-point correlations of the balanced Thue--Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue--Morse system. We finish by showing how the correlations of the Thue--Morse system with general real weights can be derived from the balanced $2$-point correlations. |
| title | Correlations of the Thue--Morse sequence |
| topic | Dynamical Systems Number Theory 37B10, 52C23 |
| url | https://arxiv.org/abs/2209.07102 |