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

Abstract

We introduce a definition of thickness in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$ with thickness $\ddot{I}„$, there is a number $N(\ddot{I}„)$ such that the set contains a translate of all sufficiently small similar copies of every set in ${\mathbb{R}}^d$ with at most $N(\ddot{I}„)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

We thank R Howat for pointing out a counterexample to Theorem 10 in the previous version. The proof fails because a combination of certain steps may lead to a degenerated case not excluded by the stated hypotheses. Under stronger assumptions one does not need to use those steps and the argument can be salvaged, see R Howats paper