Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals
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Publication:6358312
DOI10.1007/S00222-022-01171-4zbMath1526.11040arXiv2101.05797MaRDI QIDQ6358312
Publication date: 14 January 2021
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 $dgeq 1$) 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 $mathcal{L}_{d+1}$.
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