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

Kleinbock, Dmitry; Wadleigh, Nick

doi:10.1090/proc/13685

A zero-one law for improvements to Dirichlet’s Theorem
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by Dmitry Kleinbock and Nick Wadleigh PDF

Proc. Amer. Math. Soc. 146 (2018), 1833-1844 Request permission

Abstract:

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

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