An `almost all versus no' dichotomy in homogeneous dynamics and Diophantine approximation

TL;DR

This paper investigates the approximation properties of matrices within certain manifolds using homogeneous dynamics, showing that almost all matrices in a given submanifold are not very well approximable, based on orbit properties and nondivergence estimates.

Contribution

It establishes a dichotomy in homogeneous dynamics and Diophantine approximation, demonstrating that almost all matrices in a submanifold are not very well approximable, extending previous results.

Findings

Almost all matrices in the submanifold are not very well approximable.

The results are derived from quantitative nondivergence estimates.

The approach uses properties of orbits on homogeneous spaces.

Abstract

Let $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be a connected analytic submanifold in the space of $m\times n$ matrices containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.