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.
|Journal||JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY|
|Publication status||Published - 2022|
|MoE publication type||A1 Journal article-refereed|