Badly approximable grids and k-divergent lattices

Abstract

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the relationship between its measure and Diophantine properties of A. On the one hand, we give the first examples of a non-singular matrix A such that the set of badly approximable targets has full measure with respect to some non-trivial 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 A that slightly strengthens non-singularity, and show that under the assumption that A satisfies this condition, the set of badly approximable targets is a null-set with respect to any non-trivial algebraic measure on the torus. For this we use naive homogeneous dynamics, harmonic analysis, and a novel concept we refer to as mixing convergence of measures.