Equidistribution from Fractals

Abstract

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 the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. 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.

46 pages. v3: minor corrections and elaborations

References in corpus (1)

• On self-similar sets with overlaps and inverse theorems for entropy

Cited by in corpus (15)

• On self-similar sets with overlaps and inverse theorems for entropy
• Dynamics of the scenery flow and geometry of measures
• A short proof of Host's equidistribution theorem
• Conditional mixing in deterministic chaos
• Polynomial Fourier decay for fractal measures and their pushforwards
• Self-affine sets with fibered tangents
• Additive and geometric transversality of fractal sets in the integers
• Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space
• Pointwise normality and Fourier decay for self-conformal measures
• Fourier transforms of Gibbs measures for the Gauss map
• A proof of Furstenberg's conjecture on the intersections of $\times p$ and $\times q$-invariant sets
• Furstenberg's Times 2, Times 3 Conjecture (a Short Survey)
• On multifractal formalism for self-similar measures with overlaps
• On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne
• Scaling scenery of $(\times m,\times n)$ invariant measures