Trigonometric series and self-similar sets

  • Jialun Li

    Universität Zürich, Switzerland
  • Tuomas Sahlsten

    University of Manchester, UK
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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 .

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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