Hausdorff Dimension Regularity Properties and Games

TL;DR

This paper introduces a new variation of the Hausdorff dimension game that characterizes Hausdorff dimension and proves several properties and consequences, including under the axiom of determinacy, for sets in Euclidean space.

Contribution

It develops a novel variation of the Hausdorff dimension game with an unfolding property, leading to new results on dimension regularity and set structure under determinacy.

Findings

Unfolding property for the new Hausdorff dimension game.

Under AD, unions of sets with bounded Hausdorff dimension also have bounded dimension.

Every analytic set with large Hausdorff dimension contains a compact subset of nearly the same dimension.

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 $\mathsf{AD}$.