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

TL;DR

This paper defines a notion of thickness in $R^d$, providing bounds on the Hausdorff dimension of intersections of thick sets and demonstrating how thickness influences the presence of similar copies within compact sets.

Contribution

It introduces a new definition of thickness in higher dimensions and establishes key bounds and properties relating thickness to Hausdorff dimension and set intersections.

Findings

Lower bounds for Hausdorff dimension of intersections of thick sets

Existence of translations containing similar copies within thick sets

Bounds relating Hausdorff dimension and thickness

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