豪斯多夫维数的正则性与对策
Hausdorff Dimension Regularity Properties and Games
摘要：
Hausdorff $δ$ -维博弈是由Das，Fishman，Simmons和Urba{Einski}提出的，并证明了它可以刻画$δ$中具有豪斯多夫维数$\leq δ$的集合.我们介绍了这个游戏的一个变化，它也表征豪斯多夫维数，我们能够证明一个类似的基本展开性质的Banach-Mazur游戏类的展开结果。我们用它来推导出豪斯多夫维数的一些结果。我们证明了在$\mathsf{AD}$下，任何具有豪斯多夫维数$\leq δ$的集合的良序并都具有维数$\leq δ$ .我们建立了豪斯多夫维数的连续一致化结果。展开博弈还提供了一个新的证明：对于任何$δ'<δ$，每个豪斯多夫维数$\geq δ$的$\boldsymbolΣ^1_1$集合都包含维数$\geq δ'$的紧子集，并且这个结果推广到了$\mathsf{AD}$下的任意集合.
英文摘要：
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 $\mathsf{AD}$.
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