论文标题
Hausdorff维度规律性属性和游戏
Hausdorff Dimension Regularity Properties and Games
论文作者
论文摘要
Hausdorff $δ$ -Dimension Game是由DAS，Fishman，Simmons和{Urba {outa} Ski}介绍的，并显示出具有$ \ Mathbb {r}^d $具有Hausdorff Dimension $ \ leqδ$的表征。我们介绍了该游戏的一种变体，该游戏还表征了Hausdorff Dimension，我们可以证明与Banach-Mazur游戏类别的基本展开属性相似的展开结果。我们使用它来导致Hausdorff维度的许多后果。我们表明，在$ \ mathsf {ad} $下，任何井井有条的集合，每个集合都有hausdorff dimension $ \leqΔ$具有尺寸$ \leqΔ$。我们为Hausdorff维度建立了连续的均匀化结果。展开的游戏还提供了一个新的证明，即Hausdorff Dimension $ \geqδ$的每一个$ \boldsymbolς^1_1 $集包含一个紧凑的尺寸子集$ \ geqδ'$，对于任何$δ'<δ$，并且此结果概括为$ \ \ nathsf ped a ad ad ad ad a ad} $。
The Hausdorff $δ$-dimension game was introduced by Das, Fishman, Simmons and {Urba{ń}ski} and shown to characterize sets in $\mathbb{R}^d$ having Hausdorff dimension $\leq δ$. We introduce a variation of this game which also characterizes Hausdorff dimension and for which we are able to prove an unfolding result similar to the basic unfolding property for the Banach-Mazur game for category. We use this to derive a number of consequences for Hausdorff dimension. We show that under $\mathsf{AD}$ any wellordered union of sets each of which has Hausdorff dimension $\leq δ$ has dimension $\leq δ$. We establish a continuous uniformization result for Hausdorff dimension. The unfolded game also provides a new proof that every $\boldsymbolΣ^1_1$ set of Hausdorff dimension $\geq δ$ contains a compact subset of dimension $\geq δ'$ for any $δ'<δ$, and this result generalizes to arbitrary sets under $\mathsf{AD}$.
