Khintchine’s theorem on self-similar measures on the real line

Abstract: In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set. In a joint work with Timothée Bénard and Weikun He, we prove that Khintchine’s theorem holds for any self-similar measures on the real line. In particular this applies to the Hausdorff measure on the middle-thirds Cantor set. Our result generalizes the recent breakthrough work of Khalil-Luethi in dimension one. Our proof is inspired by the work of Bénard-He regarding the semisimple random walks on homogeneous spaces.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar