Random Walks, Spectral Gaps, and Khinchine Theorem on Fractals
Ergodic Theory and Dynamical Systems Seminar
12th November 2020, 4:30 pm – 5:30 pm
Online via Zoom, (if interested, please email one of the organisers to gain access to the Zoom link)
In 1984, Mahler asked how well typical points on Cantor’s set can be approximated by rational numbers. His question fits within a program, set out by himself in the 1930s, attempting to determine conditions under which subsets of R^n inherit the Diophantine properties of the ambient space. Since approximability of typical points in Euclidean space by rational points is governed by Khinchine’s classical theorem, the ultimate form of Mahler’s question asks whether an analogous zero-one law holds for fractal measures. Significant progress has been achieved in recent years, albeit, almost all known results have been of “convergence type”.
In this talk, we will discuss the first instances where a complete analogue of Khinchine’s theorem for fractal measures is obtained. The main new ingredient is an effective equidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter is established via a new technique involving the construction of S-arithmetic Markov operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. This is joint work with Manuel Luethi.