The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let ψ : R + → \psi :\mathbb {R}_+\to \mathbb {R}_+ be a non-increasing function. A real number alttext="x"> x encoding="application/x-tex">x is said to alttext="psi"> encoding="application/x-tex">\psi -Dirichlet improvable if the system stretchy="false">| q −<!-- − <mml:mi>p &gt; stretchy="false">( t stretchy="false">) and encoding="application/x-tex">\begin{equation*} |qx-p|&gt; \psi (t) \ {\text {and}} |q|&gt;t \end{equation*} has non-trivial integer solution for all large enough alttext="t"> encoding="application/x-tex">t . Denote collection of such points by alttext="upper D right-parenthesis"> D encoding="application/x-tex">D(\psi ) In this paper, we prove zero-infinity law valid dimension functions under natural non-restrictive conditions. Some consequences are laws, essentially sublinear proved Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], some non-essentially functions, but with growth condition on approximating