A localized Jarnik-Besicovitch Theorem

Details A localized Jarnik-Besicovitch Theorem A localized Jarnik-Besicovitch Theorem

2009-03-12

arxiv.org/abs/0903.2215v1

Mathematics

Number Theory

31 pages, Figures are available on our web sites

Scientific paper

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.

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