Hausdorff measure of sets of Dirichlet non-improvable affine forms

Abstract

For a decreasing real valued function ψ, a pair (A,b) of a real m×n matrix A and b∈Rm is said to be ψ-Dirichlet improvable if the system‖Aq+b−p‖m<ψ(T)and‖q‖n<T has a solution p∈Zm, q∈Zn for all sufficiently large T, where ‖⋅‖ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the ψ-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the ψ-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that b is fixed. As an application, we compute the Hausdorff dimension of the set of pairs (A,b) with uniform Diophantine exponents wˆ(A,b)≤w.