Hausdorff Dimension Regularity Properties and Games

Abstract

The Hausdorff $\delta$-dimension game was introduced by Das, Fishman, Simmons and {Urba{ń}ski} and shown to characterize sets in ${\mathbb{R}}^d$ having Hausdorff dimension $\le \delta$. 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 $\le \delta$ has dimension $\le \delta$. We establish a continuous uniformization result for Hausdorff dimension. The unfolded game also provides a new proof that every ${\mathbf{\Sigma }}_1^1$ set of Hausdorff dimension $\ge \delta$ contains a compact subset of dimension $\ge {\delta }^{\prime }$ for any ${\delta }^{\prime }<\delta$, and this result generalizes to arbitrary sets under $\mathsf{AD}$.