A localized Jarnik-Besicovitch Theorem

Abstract

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}:{\delta }_x=\delta \}$, where $\delta \ge 1$ and ${\delta }_x$ is the Diophantine approximation rate of an irrational number $x$. We go beyond the classical results by computing the Hausdorff dimension of the sets $\{x\in \mathbb{R}:{\delta }_x=f(x)\}$, 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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