Hausdorff dimension regularity properties and games

The Hausdorff δ-dimension game was introduced in [2] and shown to characterize sets in ℝ^d having Hausdorff dimension ≤ 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 AD any wellordered union of sets each of which has Hausdorff dimension ≤ δ has dimension ≤ δ. We establish a continuous uniformization result for Hausdorff dimension. The unfolded game also provides a new proof that every Σ₁¹ set of Hausdorff dimension ≥ δ contains a compact subset of dimension ≥ δ′ for any δ′ < δ, and this result generalizes to arbitrary sets under AD.