Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

TL;DR

This paper proves a version of Khintchine's Theorem for certain fractal measures, using new techniques involving spectral gaps and equidistribution on homogeneous spaces, advancing understanding of Diophantine approximation on fractals.

Contribution

It establishes the first complete analogue of Khintchine's Theorem for fractal measures generated by rational similarities, with new methods involving $S$-arithmetic operators and spectral gaps.

Findings

Khintchine's Theorem holds for specific fractal measures.

Effective equidistribution of fractal-related measures on homogeneous spaces.

New techniques involving spectral gaps and $S$-arithmetic operators.

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 $d\geq 1$) 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 $\mathcal{L}_{d+1}$.