Inventiones mathematicae

, Volume 202, Issue 1, pp 427–479

Equidistribution from fractal measures


DOI: 10.1007/s00222-014-0573-5

Cite this article as:
Hochman, M. & Shmerkin, P. Invent. math. (2015) 202: 427. doi:10.1007/s00222-014-0573-5


We give a fractal-geometric condition for a measure on \([0,1]\) to be supported on points \(x\) that are normal in base \(n\), i.e. such that \(\{n^kx\}_{k\in \mathbb {N}}\) equidistributes modulo 1. This condition is robust under \(C^1\) coordinate changes, and it applies also when \(n\) is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host’s theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.

Mathematics Subject Classification

11K16 11A63 28A80 28D05 

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsJerusalemIsrael
  2. 2.Department of Mathematics, Faculty of Engineering and Physical SciencesUniversity of SurreyGuildfordUK
  3. 3.Department of Mathematics and StatisticsTorcuato Di Tella UniversityBuenos AiresArgentina

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