Intersections of thick compact sets in $${\mathbb {R}}^d$$

Abstract

AbstractWe introduce a definition of thickness in $${\mathbb {R}}^d$$ R d and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt’s game. As an application we prove that given any compact set in $${\mathbb {R}}^d$$ R d with thickness $$\tau $$ τ , there is a number $$N(\tau )$$ N ( τ ) such that the set contains a translate of all sufficiently small similar copies of every set in $${\mathbb {R}}^d$$ R d with at most $$N(\tau )$$ N ( τ ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.