Affine thickness: Patterns and a Gap Lemma

TL;DR

This paper introduces affine thickness, a new measure for sets in Euclidean space, demonstrating its applications in game theory, set containment, and establishing a new gap lemma under specific conditions.

Contribution

It defines affine thickness as a generalization of existing thickness notions, proves its properties, and establishes a new gap lemma for affine thickness in higher dimensions.

Findings

Thick sets are winning in the matrix potential game.

Existence of homothetic copies of finite sets within thick sets.

Counter-example to the gap lemma for Falconer-Yavicoli thickness in higher dimensions.

Abstract

A new notion of thickness for subsets of $B[0,1]\subset \mathbb{R}^n$ called affine thickness is defined; this notion of thickness is a generalisation of Falconer-Yavicoli thickness and is adapted to be used in the study of certain sets with affine cut outs. Thick sets are proven to be winning for the matrix potential game introduced in (arXiv:2508.11577) and as an application we can prove that for a thick set, there exists $M\in\mathbb{N}$ depending on the thickness of the set, such that the set contains a homothetic copy of every finite set with at most $M$ elements. Additionally, the author provides a counter-example to the gap lemma in $\mathbb{R}^n$ ($n\geq 2$) for Falconer-Yavicoli thickness, stated in (Math. Z., 2022) proving this result does not hold in the generality stated. We go on to provide a gap lemma for affine thickness in $\mathbb{R}^n$ (for $n\geq 2$) under additional conditions to the classical Newhouse gap lemma.