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

Abstract

Let psi :mathbb {R}_{+}rightarrow mathbb {R}_{+} be a non-increasing function. A pair (A,{textbf{b}}), where A is a real mtimes n matrix and {textbf{b}}in mathbb {R}^{m}, is said to be psi -Dirichlet improvable, if the system ‖Aq+b-p‖m<ψ(T),‖q‖n<T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert A{\ extbf{q}} +{\ extbf{b}}-{\ extbf{p}}\\Vert ^m<\\psi (T), \\quad \\Vert {\ extbf{q}}\\Vert ^n<T \\end{aligned}$$\\end{document}is solvable in {textbf{p}}in mathbb {Z}^{m},{textbf{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 (2022) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff f-measure version of Kim–Kim (2022) results for psi -Dirichlet non-improvable sets.