Patterns in thick compact sets

TL;DR

This paper establishes a link between Newhouse thickness and pattern presence in compact sets, using a modified Schmidt's game to provide a practical condition for detecting patterns, especially in Cantor sets.

Contribution

It introduces a new connection between thickness and patterns via a variant of Schmidt's game, offering a checkable condition for pattern existence in compact sets.

Findings

Provides an explicit condition for pattern presence in compact sets.

Connects Newhouse thickness with pattern detection.

Applies to Cantor sets specifically.

Abstract

We introduce a connection between Newhouse thickness and patterns through a variant of Schmidt's game introduced by Broderick, Fishman and Simmons. This yields an explicit, robust and checkable condition that ensures the presence of patterns in compact sets, in particular in Cantor sets.