Trigonometric series and self-similar sets

Jialun Li, Tuomas Sahlsten*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. The rate of $\widehat{\mu}(\xi) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for stopping times of random walks on $\mathbb{R}$.
Original language English JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY https://doi.org/10.4171/JEMS/1102 Published - 17 Aug 2021 A1 Journal article-refereed

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