Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem, that is if 
the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$
has a non-trivial integer solution for all large enough $t$.  In this talk, I will explain that the Hausdorff measure of the set of  $\psi$-Dirichlet non-improvable numbers obeys a zero-infinity law for a large class of dimension functions. 

Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.