Shrinking targets problems for flows on homogeneous spaces

TL;DR

This paper investigates how certain flows on homogeneous spaces behave with respect to shrinking targets, establishing new logarithm laws for cusp excursions and covering both diagonalizable and unipotent flows.

Contribution

It introduces general results for shrinking target problems on homogeneous spaces, including new logarithm laws for unipotent flows' cusp excursions.

Findings

Established logarithm laws for cusp excursions of unipotent flows.

Applied results to both diagonalizable and unipotent flows.

Covered very general families of shrinking targets.

Abstract

We study shrinking targets problems for discrete time flows on a homogenous space $\Gamma\backslash G$ with $G$ a semisimple group and $\Gamma$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.