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We give an integrability condition on a function $\psi$ guaranteeing that for almost all (or almost no) $x\in\mathbb{R}$, the system $|qx-p|\leq \psi(t)$, $|q|<t$ is solvable in $p\in \mathbb{Z}$, $q\in \mathbb{Z}\smallsetminus \{0\}$ for sufficiently large $t$. Along the way, we characterize such $x$ in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.

A zero-one law for improvements to Dirichlet’s Theorem