Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $SL_2(\mathbb{R})$

TL;DR

This paper establishes that the set of points with bounded orbits under certain flows on finite volume quotients of products of SL_2(R) is large in a measure-theoretic sense, using number theory and dynamical systems techniques.

Contribution

It proves a number field analogue of Schmidt's conjecture and shows that bounded orbits form a hyperplane absolute winning set on these quotients.

Findings

Proves a number field analogue of Schmidt's conjecture.

Shows bounded orbits form a hyperplane absolute winning set.

Connects Diophantine approximation with dynamical properties of flows.

Abstract

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let $G := SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) $ and $\Gamma$ be a lattice in $G$. We show that the set of points on $G/\Gamma$ whose forward orbits under a one parameter Ad-semisimple subsemigroup of $G$ are bounded, form a hyperplane absolute winning set.