Let be a self-similar set on associated to contractions , , for some finite , such that is not a singleton. We prove that if is irrational for some , then is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of . No separation conditions are assumed on . We establish our result by showing that every self-similar measure on is a Rajchman measure: the Fourier transform as . The rate of is also shown to be logarithmic if is diophantine for some . The proof is based on quantitative renewal theorems for stopping times of random walks on .
Cite this article
Jialun Li, Tuomas Sahlsten, Trigonometric series and self-similar sets. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368DOI 10.4171/JEMS/1102