THE GENERALISED HAUSDORFF MEASURE OF SETS OF DIRICHLET NON-IMPROVABLE NUMBERS

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