Absolute continuity of self-similar measures, their projections and convolutions

Pablo Shmerkin, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

33 Scopus citations

Abstract

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.

Original languageEnglish
Pages (from-to)5125-5151
Number of pages27
JournalTransactions of the American Mathematical Society
Volume368
Issue number7
DOIs
StatePublished - Jul 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 American Mathematical Society.

Keywords

  • Absolute continuity
  • Convolutions
  • Hausdorff dimension
  • Self-similar measures

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