Badly approximable grids and 𝒌-divergent lattices

Abstract

Let 𝐴 ∈ Mat𝑚×𝑛(R) be a matrix. In this paper, we investigate the set Bad𝐴 ⊂ 𝕋𝑚 of badly approximable targets for 𝐴, where 𝕋𝑚 is the 𝑚-torus. It is well known that Bad𝐴 is a winning set for Schmidt’s game and hence is a dense subset of full Hausdorff dimension. We investi- gate the relationship between the measure of Bad𝐴 and Diophantine properties of 𝐴. On the one hand, we give the first examples of a nonsingular 𝐴 such that Bad𝐴 has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference the- orems 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 𝐴 that slightly strengthens nonsingularity, and show that under the assumption that 𝐴 satisfies this condition, Bad𝐴 is a null-set with respect to any nontrivial algebraic mea- sure 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.