Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $S{L}_2(\mathbb{R})$

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 $G:=S{L}_2(\mathbb{R})\times \cdots \times S{L}_2(\mathbb{R})$ 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.

Changes following referee report. This is the final version. To appear in Advances in Mathematics

References in corpus (3)

• Badly approximable points on manifolds
• Badziahin-Pollington-Velani's theorem and Schmidt's game
• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

Cited by in corpus (1)

• Cylinder absolute games on solenoids