A localized Jarnik-Besicovitch Theorem

Abstract

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}: δ_x = δ\}$, where $δ\geq 1$ and $δ_x$ is the Diophantine approximation rate of an irrational number $x$. We go beyond the classical results by computing the Hausdorff dimension of the sets $\{x\in\mathbb{R}: δ_x =f(x)\}$, where $f$ is a continuous function. Our theorem applies to the study of the approximation rates by various approximation families. It also applies to functions $f$ which are continuous outside a set of prescribed Hausdorff dimension.

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