Absolute continuity of non-homogeneous self-similar measures

Santiago Saglietti, Pablo Shmerkin, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S.-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.

Original languageEnglish
Pages (from-to)60-110
Number of pages51
JournalAdvances in Mathematics
Volume335
DOIs
StatePublished - 7 Sep 2018

Bibliographical note

Publisher Copyright:
© 2018

Keywords

  • Absolute continuity
  • Convolutions
  • Random measures
  • Self-similar measures

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