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

TL;DR

This work introduces a higher-dimensional thickness for compact sets in $\mathbb{R}^d$, enabling quantitative control over intersections of thick sets. By developing a Schmidt-game–type framework, it derives lower bounds on the Hausdorff dimension of intersections and proves a higher-dimensional Gap Lemma for certain linked gaps. The main results show that sufficiently thick families of compact sets have nonempty intersections and admit explicit dimension bounds, which in turn yield pattern-embedding results: thick sets contain homothetic copies of finite configurations up to a computable size $N(\tau)$. These findings connect thickness to dimension and pattern presence, with concrete applications to fractal constructions such as Sierpiński carpets/sponges and to the study of patterns in zero-measure sets. The work thus provides a robust, computable thickness–based criterion for nontrivial intersections and pattern containment 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 $τ$, there is a number $N(τ)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(τ)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

Theorems & Definitions (29)

• Definition 1: Thickness in $\mathbb{R}$
• Theorem 2: Newhouse's Gap Lemma
• Definition 3: Thickness in $\mathbb{R}^d$
• Definition 5
• Theorem 6: Intersection of compact sets in $\mathbb{R}^d$
• Theorem 7
• Definition 8
• Proposition 9
• proof
• Definition 10