Equidistribution from fractal measures

Michael Hochman, Pablo Shmerkin

Research output: Contribution to journalArticlepeer-review


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 (Formula presented.) equidistributes modulo 1. This condition is robust under C1 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.

Original languageAmerican English
Pages (from-to)427-479
Number of pages53
JournalInventiones Mathematicae
Issue number1
StatePublished - 1 Oct 2015


  • 11A63
  • 11K16
  • 28A80
  • 28D05

All Science Journal Classification (ASJC) codes

  • General Mathematics


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