Random walks on homogeneous spaces, Spectral Gaps, and Khintchine's theorem on fractals

Vortrag von Dr. Manuel Lüthi

Datum: 08.11.21  Zeit: 15.00 - 16.00  Raum: Y27H28

Khintchine's theorem in Diophantine approximation gives a zero one law describing the approximability of typical points by rational points. In 1984, Mahler asked how well points on the middle third Cantor set can be approximated. His question fits into an attempt to determine conditions under which subsets of Euclidean space inherit the Diophantine properties of the ambient space. I will discuss a complete analogue of the theorem of Khintchine for certain fractal measures which was recently obtained in collaboration with Osama Khalil. Our results hold for fractals generated by rational similarities of Euclidean space that have sufficiently small Hausdorff co-dimension. The main ingredient to the proof is an effective equidistribution theorem for associated fractal measures on the space of unimodular lattices. The latter is established using a spectral gap property of a type of Markov operators associated with an S-arithmetic random walk related to the generating similarities.