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 thirds 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 ninths Cantor set"; i.e. the set of numbers whose base 9 expansions miss a single digit.The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space 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 L d+1 .

The Divergence Borel-Cantelli Lemma revisited

“…The divergence part is also known as the second Borel-Cantelli Lemma and it naturally shows up (in some form) in the proof of the notorious Duffin-Schaeffer Conjecture [17] recently given by Koukoulopoulos & Maynard [25] and its higher dimensional generalisation proved two decades earlier by Pollington & Vaughan [30]. Indeed, the divergence Borel-Cantelli Lemma is very much at the heart of numerous other recent advances on topical problems in metric number theory, such as those in the theory of multiplicative and inhomogeneous Diophantine approximation and Diophantine approximation on manifolds and more generally on fractals, see for example [1,5,12,13,14,15,23,32,33,36]. In a nutshell, our goal it is to revisit the Borel-Cantelli Lemma and to establish both sufficient and necessary conditions that guarantee either positive or full measure.…”

Approximating elements of the middle third Cantor set with dyadic rationals

“…These two questions have generated a substantial amount of research (see [2,3,4,5,7,10,11,12,21,23,24,29,31,32,33,35] and the references therein). We do not attempt to give an exhaustive overview of research in this area.…”

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

“…Shah's results have been generalized and strengthened in various directions [33,34,37,17]. In a recent breakthrough of Khalil and Luethi [14], the authors refined (1.14) (for the case when n = 1) by replacing Leb with a certain fractal measure, from which they deduced a complete analogue of Khintchine's theorem with respect to this fractal measure.…”