Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

Abstract

Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a nonincreasing function. A pair $(A,\mathbf{b}),$ where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in {\mathbb{R}}^m,$ is said to be $\psi$-Dirichlet improvable, if the system $$\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}{\Vert }^m<\psi (T),\Vert \mathbf{q}{\Vert }^n<T$$ is solvable in $\mathbf{p}\in {\mathbb{Z}}^m,$ $\mathbf{q}\in {\mathbb{Z}}^n$ for all sufficiently large $T$ where $\Vert \cdot \Vert$ denotes the supremum norm. For $\psi$-Dirichlet non-improvable sets, Kleinbock--Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim--Kim (2021) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff $f$-measure version of Kim--Kim (2021) results for $\psi$-Dirichlet non-improvable sets.