  Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real
number $x$ is said to be $\psi$-Dirichlet improvable if 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 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 sub-linear dimension functions proved by
Hussain-Kleinbock-Wadleigh-Wang (2018), for some non-essentially sub-linear
dimension functions, and for all dimension functions but with a growth
condition on the approximating function.
