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arXiv · 2401.01120

Van der Corput and metric theorems for geometric progressions for self-similar measures

Amir Algom·Yuanyang Chang·Meng Wu·Yu-Liang Wu
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Abstract

We prove a van der Corput lemma for non-atomic self-similar measures $\mu$. 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 $\mu$-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.

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Amir Algom, Yuanyang Chang, Meng Wu, Yu-Liang Wu. 2024-01-02. Van der Corput and metric theorems for geometric progressions for self-similar measures. https://arxiv.org/abs/2401.01120

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