Thickness and a gap lemma in $\mathbb{R}^d$

TL;DR

This paper introduces a new notion of thickness in rd4d, proves a Gap Lemma type result, and explores properties of thick sets including dimension bounds and pattern presence.

Contribution

It defines a useful concept of thickness in rd4d for totally disconnected sets and establishes several fundamental properties and results related to these sets.

Findings

Proves a Gap Lemma type result for thick sets.

Establishes a lower bound for the Hausdorff dimension of intersections of thick sets.

Shows the presence of large patterns in thick sets.

Abstract

We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets (for this definition of thickness) with winning sets, give a lower bound for the Hausdorff dimension of the intersection of countably many of them, a result guaranteeing the presence of large patterns, and lower bounds for the Hausdorff dimension of a set in relationship with its thickness.