A survey on Newhouse thickness, fractal intersections and patterns

TL;DR

This survey explores the concept of thickness in sets, its role in Cantor set intersections, and its connections with Schmidt Games and patterns, mainly focusing on the real line and extending to higher dimensions.

Contribution

It introduces a new notion of size called thickness, reviews its applications in intersections and patterns, and discusses its relation to Schmidt Games across dimensions.

Findings

Thickness guarantees Cantor set intersections via the Gap Lemma.

Connections established between Thickness, Schmidt Games, and Patterns.

Extension of concepts from the real line to higher dimensions.

Abstract

In this article, we introduce a notion of size for sets called thickness that can be used to guarantee that two Cantor sets intersect (the Gap Lemma), and show a connection among Thickness, Schmidt Games and Patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions.