JournalsjemsVol. 24, No. 1pp. 341–368

Trigonometric series and self-similar sets

  • Jialun Li

    Universität Zürich, Switzerland
  • Tuomas Sahlsten

    University of Manchester, UK
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Let F{F} be a self-similar set on R\mathbb{R} associated to contractions fj(x)=rjx+bj{f_j(x) = r_j x + b_j}, jA{j \in \mathcal{A}}, for some finite A\mathcal{A}, such that F{F} is not a singleton. We prove that if logri/logrj{\log r_i / {\log r_j}} is irrational for some ij{i \neq j}, then F{F} is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of F{F}. No separation conditions are assumed on F{F}. We establish our result by showing that every self-similar measure μ{\mu} on F{F} is a Rajchman measure: the Fourier transform μ^(ξ)0{\widehat{\mu}(\xi) \to 0} as ξ{|\xi| \to \infty}. The rate of μ^(ξ)0{\widehat{\mu}(\xi) \to 0} is also shown to be logarithmic if logri/logrj{\log r_i / {\log r_j}} is diophantine for some ij{i \neq j}. The proof is based on quantitative renewal theorems for stopping times of random walks on R\mathbb{R}.

Cite this article

Jialun Li, Tuomas Sahlsten, Trigonometric series and self-similar sets. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368

DOI 10.4171/JEMS/1102