Let
$\unicode[STIX]{x1D713}:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$
be a non-increasing function. A real number
$x$
is said to be
$\unicode[STIX]{x1D713}$
-Dirichlet improvable if it admits an improvement to Dirichlet’s theorem in the following sense: the system
$$\begin{eqnarray}|qx-p|<\unicode[STIX]{x1D713}(t)\quad \text{and}\quad |q|<t\end{eqnarray}$$
has a non-trivial integer solution for all large enough
$t$
. Denote the collection of such points by
$D(\unicode[STIX]{x1D713})$
. In this paper we prove that the Hausdorff measure of the complement
$D(\unicode[STIX]{x1D713})^{c}$
(the set of
$\unicode[STIX]{x1D713}$
-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 and Wadleigh [A zero-one law for improvements to Dirichlet’s theorem.
Proc. Amer. Math. Soc.
146 (2018), 1833–1844], our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.