Hausdorff Dimension Regularity Properties and Games

Abstract

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