论文标题
确切的近似顺序和分布良好的集合
Exact approximation order and well-distributed sets
论文作者
论文摘要
我们证明，对于任何适当的度量空间$ x $和功能$ψ：（0，\ infty）\ to（0，\ infty）$从适当的近似函数中，套件的$w_ψ（q）的hausdorff尺寸，所有点的$ n $ n $ - plopproxim_-wellib $ quim_- $ quy_的$ q $ q q oft seet seet quits $ q q \ q \ q q，以及该集合的$ q q \ q \ q \ q Q \ q \ q Q \ Q Q \ Q \ Q \ Q \ Q Q，Q \ Q Q \ q \ Q Q \ q \ Q Q \ q \ Q Q \ q \ Q \恰好是$ n $ - $ q $，conciend。这是在一般环境中的答案，这是贝雷斯内维奇 - 迪金森·维拉尼（Beresnevich-Dickinson-Velani）在理性近似的情况下，在这种情况下，Bugeaud使用REAL的持续扩展来回答。我们的主要结果特别适用于近似作用在双曲度度空间边界上的宽类异构体的固定点的轨道。
We prove that for any proper metric space $X$ and a function $ψ:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_ψ(Q)$ of all points $ψ$-well-approximable by a well-distributed subset $Q\subset X$, and the set $E_ψ(Q)$ of points that are exactly $ψ$-approximable by $Q$, coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.
