Singularity versus exact overlaps for self-similar measures

Károly Simon, Lajos Vágó, Alexander Iosevich

Research output: Contribution to journalArticle


In this note we present some one-parameter families of homogeneous self-similar measures on the line such that • the similarity dimension is greater than 1 for all parameters and • the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-α projections of the natural measure of the Sierpiński carpet. We present more general one-parameter families of self-similar measures vα, such that the set of parameters α for which vα is singular is a dense G δ set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.

Original languageEnglish
Pages (from-to)1971-1986
Number of pages16
JournalProceedings of the American Mathematical Society
Issue number5
Publication statusPublished - May 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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