Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation I
Speaker: Osama Khalil, University of Utah
Date: November 3, 2021
Time: 17:30:00 Hours
Venue: AG-77
Abstract: Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational ones. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at determining conditions under which Diophantine properties of Euclidean space are inherited by its various subsets of interest.
Title of Talk: Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation I
Speaker: Osama Khalil, University of Utah
Date: November 3, 2021
Time: 17:30:00 Hours
Venue: AG-77
Abstract: Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational ones. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at determining conditions under which Diophantine properties of Euclidean space are inherited by its various subsets of interest.
Over the course of two lectures, we will discuss recent joint work with Manuel Luethi yielding the first complete analogue of Khintchine’s Theorem for certain self-similar fractal measures. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices. To prove the latter, we associate to such fractals certain $p$-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. We will also discuss some unexpected difficulties in the deduction of the divergence part of the analogue of Khintchine's Theorem as manifestations of certain subconvexity problems.