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

TL;DR

This paper establishes a zero-one law for the measure of real numbers satisfying certain Diophantine approximation conditions based on an integrability criterion for the function cpsi, and explores the sharpness of Dirichlet's theorem with higher-dimensional extensions.

Contribution

It introduces an integrability condition on cpsi that determines almost all or almost no real numbers satisfy specific Diophantine inequalities, and characterizes such numbers via continued fractions.

Findings

Characterization of real numbers based on continued fraction growth

An integrability criterion for cpsi guaranteeing measure-zero or measure-one sets

Demonstration of the sharpness of Dirichlet's Approximation 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 $|qx-p|\leq \psi(t)$, $|q|<t$ 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.