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Affine thickness: Patterns and a Gap Lemma

Richard A. Howat

TL;DR

This work introduces affine thickness $\tau_A(C)$ for subsets of $B[0,1]$ using a diagonal matrix $A$ to generalize Falconer–Yavicoli thickness and to study sets with affine cutouts. It shows that a direct extension of Newhouse's gap lemma to higher dimensions fails for Falconer–Yavicoli thickness, but an affine gap lemma can hold under BG linkedness and strong refinability assumptions, enabling robust intersection results. The paper proves that thick sets are winning in the matrix potential game and derives consequences for pattern formation and intersections, including lower bounds on pattern dimensions and stability under countable intersections. It culminates with concrete examples via Self-Affine Sierpinski carpets, where explicit choices of $\underline{r}$ yield large finite-pattern capacities, demonstrating the practical impact for fractal geometry and affine thickness methods.

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.

Affine thickness: Patterns and a Gap Lemma

TL;DR

This work introduces affine thickness for subsets of using a diagonal matrix to generalize Falconer–Yavicoli thickness and to study sets with affine cutouts. It shows that a direct extension of Newhouse's gap lemma to higher dimensions fails for Falconer–Yavicoli thickness, but an affine gap lemma can hold under BG linkedness and strong refinability assumptions, enabling robust intersection results. The paper proves that thick sets are winning in the matrix potential game and derives consequences for pattern formation and intersections, including lower bounds on pattern dimensions and stability under countable intersections. It culminates with concrete examples via Self-Affine Sierpinski carpets, where explicit choices of yield large finite-pattern capacities, demonstrating the practical impact for fractal geometry and affine thickness methods.

Abstract

A new notion of thickness for subsets of 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 depending on the thickness of the set, such that the set contains a homothetic copy of every finite set with at most elements. Additionally, the author provides a counter-example to the gap lemma in () 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 (for ) under additional conditions to the classical Newhouse gap lemma.
Paper Structure (7 sections, 19 theorems, 41 equations, 1 figure)

This paper contains 7 sections, 19 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An Affine Gap Lemma in $\mathbb{R}^n$
  3. Gap lemma for BG linked pairs of sets
  4. A Condition for strong refinability
  5. Affine thickness and the matrix potential game
  6. Intersections and pattern properties in sets with large thickness
  7. An example: Self-Affine Sierpinski carpets

Key Result

Proposition 1

Consider the square or Euclidean metric on $\mathbb{R}^n$ and $n\geq 2$. Let $A$ be a diagonal matrix with entries $\beta_{11},\dots,\beta_{nn}\in (0,1)$. There exists compact sets $C_1,C_2\subseteq B[0,1]$ such that neither of them is contained in a gap of the other, $\tau_A(C_1)+\tau_A(C_2)>0$ and

Figures (1)

  • Figure 1: The sets $C_1$ and $C_2$ both individually and together contained in $B[0,1]$ with circles denoting the singletons at $x$ and $-x$.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition
  • Corollary
  • Theorem : Affine Gap Lemma
  • Theorem
  • Theorem
  • ...and 30 more