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