Speaker
Timothée Bénard, Université Sorbonne Paris Nord
Abstract
The Khintchine theorem is one of the cornerstones of Diophantine approximation. It predicts how well Lebesgue-typical real numbers can be approximated by rationals. In this mini-course, I will explain how to extend this theorem to typical points chosen according to the middle-thirds Cantor measure. This result, obtained jointly with Weikun He and Han Zhang, answers a question of K. Mahler from the 1980s regarding Diophantine approximation on fractals. The proof relies on the effective equidistribution of an associated random walk on the homogeneous space $\mathrm{SL}{2}(\mathbb{R})/\mathrm{SL}{2}(\mathbb{Z})$, which in turn exploits a multislicing extension of Bourgain’s projection theorem.
Sessions will be organized as follows:
Introduction to the fractal Khintchine theorem and reduction to random walks.
From a point to positive dimension, from high dimension to equidistribution.
From positive dimension to high dimension: the multislicing argument.