  For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a
real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be
$\psi$-Dirichlet improvable if the system
$$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m <
\psi(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution
$\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently
large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh
(2019) established an integrability criterion for the Lebesgue measure of the
$\psi$-Dirichlet non-improvable set. In this paper, we prove a similar
criterion for the Hausdorff measure of the $\psi$-Dirichlet non-improvable set.
Also, we extend this result to the singly metric case that $\mathbf{b}$ is
fixed. As an application, we compute the Hausdorff dimension of the set of
pairs $(A,\mathbf{b})$ with uniform Diophantine exponents
$\widehat{w}(A,\mathbf{b})\leq w$.
