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

Abstract

AbstractLet $$\psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$$ ψ : R + → R + be a non-increasing function. A pair $$(A,{\textbf{b}}),$$ ( A , b ) , where A is a real $$m\times n$$ m × n matrix and $${\textbf{b}}\in \mathbb {R}^{m},$$ b ∈ R m , is said to be $$\psi $$ ψ -Dirichlet improvable, if the system $$\begin{aligned} \Vert A{\textbf{q}} +{\textbf{b}}-{\textbf{p}}\Vert ^m<\psi (T), \quad \Vert {\textbf{q}}\Vert ^n<T \end{aligned}$$ ‖ A q + b - p ‖ m < ψ ( T ) , ‖ q ‖ n < T is solvable in $${\textbf{p}}\in \mathbb {Z}^{m},$$ p ∈ Z m , $${\textbf{q}}\in \mathbb {Z}^{n}$$ q ∈ Z n for all sufficiently large T where $$\Vert \cdot \Vert $$ ‖ · ‖ denotes the supremum norm. For $$\psi $$ ψ -Dirichlet non-improvable sets, Kleinbock–Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim–Kim (2022) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff f-measure version of Kim–Kim (2022) results for $$\psi $$ ψ -Dirichlet non-improvable sets.