Van der Corput and metric theorems for geometric progressions for self-similar measures
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914627078586368 |
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| author | Algom, Amir Chang, Yuanyang Wu, Meng Wu, Yu-Liang |
| author_facet | Algom, Amir Chang, Yuanyang Wu, Meng Wu, Yu-Liang |
| contents | We prove a van der Corput lemma for non-atomic self-similar measures $μ$. As an application, we show that the correlations of all finite orders of $( x^n \mod 1 )_{n\geq 1}$ converge to the Poissonian model for $μ$-a.e. $x$, assuming $x>1$. We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Van der Corput and metric theorems for geometric progressions for self-similar measures Algom, Amir Chang, Yuanyang Wu, Meng Wu, Yu-Liang Dynamical Systems Classical Analysis and ODEs Number Theory We prove a van der Corput lemma for non-atomic self-similar measures $μ$. As an application, we show that the correlations of all finite orders of $( x^n \mod 1 )_{n\geq 1}$ converge to the Poissonian model for $μ$-a.e. $x$, assuming $x>1$. We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay. |
| title | Van der Corput and metric theorems for geometric progressions for self-similar measures |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2401.01120 |