A zero-one Law for improvements to Dirichlet's Theorem

Abstract

We give an integrability condition on a function $\psi$ guaranteeing that for almost all (or almost no) $x\in \mathbb{R}$, the system $|qx-p|\le \psi (t)$, $|q|<t$ is solvable in $p\in \mathbb{Z}$, $q\in \mathbb{Z}\setminus \{0\}$ 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.

arXiv comment12 pages; minor corrections made, Corollary 3.7 added to the latest version