  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 in the following sense: the system $$|qx-p|<
\, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for
all large enough $t$. Denote the collection of such points by $D(\psi)$. In
this paper, we prove that the Hausdorff measure of the complement $D(\psi)^c$
(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.
