Dmitry Kleinbock: Shrinking targets on homogeneous spaces and improving [...] (NTWS 120)

Title: Shrinking targets on homogeneous spaces and improving Dirichlet's Theorem

Abstract: Let $\psi$ be a decreasing function defined on all large positive real numbers. We say that a real $m \times n$ matrix $Y$ is "$\psi$-Dirichlet" if for every sufficiently large real number $T$ there exist non-trivial integer vectors $(p,q)$ satisfying $\|Yq-p\|^m \less \psi(T)$ and $\|q\|^n \less T$ (where $\|\cdot\|$ denotes the supremum norm on vectors). This generalizes the property of $Y$ being "Dirichlet improvable" which has been studied by several people, starting with Davenport and Schmidt in 1969. I will present results giving sufficient conditions on $\psi$ to ensure that the set of $\psi$-Dirichlet matrices has zero (resp., full) measure. If time allows I will mention a geometric generalization of the set-up, where the supremum norm is replaced by an arbitrary norm. Joint work with Anurag Rao, Andreas Strombergsson, Nick Wadleigh and Shuchweng Yu.

Original air date:
Thursday, March 10, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 11, 2022 (12am CST, 3am AEDT, 5am NZDT)