Trigonometric series and self-similar sets

Jialun Li, Tuomas Sahlsten*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)

Abstract

Let F be a self-similar set on R associated to contractions fj (x) = r(j) x + bj , j is an element of A, for some finite A, such that F is not a singleton. We prove that if log ri =log rj is irrational for some i # 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 mu(xi) -> infinity as ItI ! oo. The rate of il,(t) ! 0 is also shown to be logarithmic if log ri =log rj is diophantine for some i # j. The proof is based on quantitative renewal theorems for stopping times of random walks on R.

Original languageEnglish
Pages (from-to)341-368
JournalJOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
Volume24
Issue number1
DOIs
Publication statusPublished - 2022
MoE publication typeA1 Journal article-refereed

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