Inventiones mathematicae

, Volume 202, Issue 1, pp 427–479 | Cite as

Equidistribution from fractal measures

  • Michael HochmanEmail author
  • Pablo Shmerkin


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 



Part of this work was carried out while M.H. was visiting at the Theory group at Microsoft Research (Redmond); many thanks to the members of the group for their hospitality and support.


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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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