一个局部化的Jarnik-Besicovitch定理
A localized Jarnik-Besicovitch Theorem
摘要：
丢番图逼近中的基本问题与$\{x\in \mathbb{R}: δ_x = δ\}$形式的集合的豪斯多夫维数有关，其中$δ\geq 1$和$δ_x$是无理数$x$的丢番图逼近率。通过计算集合$\{x\in\mathbb{R}: δ_x =f(x)\}$的豪斯多夫维数，我们超越了经典的结果，其中$f$是一个连续函数.我们的定理适用于各种逼近族的逼近率的研究。它也适用于函数$f$这是连续的一组规定的豪斯多夫维数。
英文摘要：
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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