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

By Dmitry Kleinbock, Nick Wadleigh. 2016

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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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b84f003aa4720dd6fb39293e0daf5d76f4bc49df

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

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Hausdorff measure of sets of Dirichlet non-improvable numbers (2017, background): “Together with the Lebesgue measure-theoretic results established by Kleinbock and Wadleigh [A zero-one law for improvements to Dirichlet’s theorem.”