Hausdorff Dimension Regularity Properties and Games

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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