Speaker: Osama Khalil
3 June, 2020 14:00 UTC
Abstract. 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 $\mathbb{R}^d$ inherit the Diophantine properties of the ambient space. Since the approximability of typical points in Euclidean space by rational points is governed by Khintchine’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 class of fractals for which our results hold includes those generated by rational similarities of $\mathbb{R}^d$ and having sufficiently small Hausdorff co-dimension. 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 in progress with Manuel Luethi.