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

• 成果类型: Article
• 署名作者: Khalil, Osama; Luethi, Manuel
• 署名单位: Utah System of Higher Education; University of Utah; Tel Aviv University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01171-4
• 发表日期: 2023
• 页码: 713-831
• 关键词: diophantine approximation; stationary measures; homogeneous spaces; rational-points; unitary representations; invariant subsets; semisimple groups; horocycle flow; planar curves; equidistribution

摘要

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 R-d (for any d >= 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 Ld+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 Ld+1.