A localized Jarnik-Besicovitch Theorem

TL;DR

This paper extends classical Diophantine approximation results by computing the Hausdorff dimension of sets where the approximation rate varies with a continuous function, broadening understanding of approximation behaviors.

Contribution

It introduces a localized version of the Jarnik-Besicovitch theorem, allowing the Hausdorff dimension calculation for sets defined by variable approximation rates.

Findings

Calculates Hausdorff dimension for sets with variable approximation rates

Applies to functions continuous outside sets of prescribed Hausdorff dimension

Generalizes classical results to broader classes of functions

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 $\delta \geq 1$ 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.