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

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