Trigonometric series and selfsimilar sets
Abstract
Let $F$ be a selfsimilar 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 selfsimilar 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 $\mathbb{R}$.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.00426
 Bibcode:
 2019arXiv190200426L
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 Mathematics  Spectral Theory;
 42A20 (Primary);
 42A38;
 37C45;
 28A80;
 60K05 (Secondary)
 EPrint:
 24 pages, 1 figure, v3: added details on the renewal theorem side, revised version. To appear in JEMS