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

Let ψ : R+ → R+ be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). 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.