Random walks, spectral gaps, and Khintchine’s theorem on fractals

Abstract

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor’s middle 1/3 set. We obtain the first instances where a complete analogue of Khintchine’s Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of \({\mathbb {R}}^d\) (for any \(d\ge 1\)) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler’s problem is the Hausdorff measure on the “middle \(1/5\) Cantor set”; i.e. the set of numbers whose base \(5\) expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space \({\mathcal {L}}_{d+1}\) of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of S-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on \({\mathcal {L}}_{d+1}\).