Badly approximable grids and -divergent lattices

Abstract

Let $$ be a matrix. In this paper, we investigate the set $$ of badly approximable targets for $$, where $$ is the $$-torus. It is well known that $$ is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of $$ and Diophantine properties of $$. On the one hand, we give the first examples of a nonsingular $$ such that $$ has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on $$ that slightly strengthens nonsingularity, and show that under the assumption that $$ satisfies this condition, $$ is a null-set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.