# Trigonometric series and self-similar sets

## TL;DR

This paper demonstrates that certain self-similar sets in one dimension are sets of multiplicity, meaning trigonometric series are not unique outside these sets, by showing associated measures have Fourier transforms tending to zero.

## Contribution

It proves that self-similar sets with irrational log-ratio contractions are sets of multiplicity, establishing Fourier decay for measures on these sets without separation conditions.

## Key findings

- Self-similar sets with irrational log ratios are sets of multiplicity.
- Self-similar measures on these sets are Rajchman measures.
- Fourier transform decay rate is logarithmic under diophantine conditions.

## 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 $\mathbb{R}$.