Let
ψ
:
R
+
→
R
+
\psi :\mathbb {R}_+\to \mathbb {R}_+
be a non-increasing function. A real number
x
x
is said to be
ψ
\psi
-Dirichlet improvable if the system
|
q
x
−
p
|
>
ψ
(
t
)
and
|
q
|
>
t
\begin{equation*} |qx-p|> \psi (t) \ \ {\text {and}} \ \ |q|>t \end{equation*}
has a non-trivial integer solution for all large enough
t
t
. Denote the collection of such points by
D
(
ψ
)
D(\psi )
. In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.