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Triangles in the Plane and arithmetic progressions in thick compact subsets of $\mathbb{R}^d$

Samantha Sandberg-Clark, Krystal Taylor

Abstract

This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb{R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb{R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided the Newhouse thickness of $A$ is at least $1$. Moreover, we prove that compact sets $A\subset \mathbb{R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of $C$ is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.These are among the first results in the literature to give explicit criteria for the occurrence of three-point configurations in the plane.

Triangles in the Plane and arithmetic progressions in thick compact subsets of $\mathbb{R}^d$

Abstract

This article focuses on the occurrence of 3-point configurations in subsets of of sufficient thickness. We prove that a compact set contains a similar copy of any linear -point configuration (such as a -point arithmetic progression) provided satisfies a mild Yavicoli-thickness condition and an -uniformity condition for ; or, when , the result holds provided the Newhouse thickness of is at least . Moreover, we prove that compact sets contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided satisfies a mild Yavicoli-thickness condition and an -uniformity condition. Further, contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of is at least . These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.These are among the first results in the literature to give explicit criteria for the occurrence of three-point configurations in the plane.

Paper Structure

This paper contains 21 sections, 20 theorems, 180 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Newhouse thickness
  3. The Gap Lemma
  4. Acknowledgment
  5. Main Results
  6. 3-point configurations in $\mathbb{R}$ & Triangles in the plane part I
  7. 3-point configurations in $\mathbb{R}^d$ & Triangles in the plane part II
  8. Organization
  9. Yavicoli Thickness in $\mathbb{R}^d$
  10. Computing the thickness of a subset
  11. Proof of Main Results that use Newhouse thickness
  12. Proof of Proposition \ref{['prop:assymetric progressions updated']}
  13. Proof of Theorem \ref{['thm: triangles Cartesian']}
  14. Proof of Main Results that use Yavicoli thickness
  15. Setup and Table of notation for the proofs of Theorems \ref{['thm: higher dim convex combo']} and \ref{['thm: triangles no Cartesian']}
  16. ...and 6 more sections

Key Result

Proposition 1.2

Let $C\subset\mathbb{R}$ be a compact set with $\tau(C)\geq 1$. Then $C$ contains an arithmetic progression of length $3$.

Figures (8)

  • Figure 1: We see that $C\times C$ contains an equilateral triangle by combining two facts: (i) $C$ contains an arithmetic progression $\mathcal{A} = \{x, x+t, x+2t\}$, where $t>0$ can be taken arbitrarily small; (ii) the distance set $\Delta(C)$ contains an interval $[0,\ell]$ for some $\ell>0$.
  • Figure 2: First two iterations of off-center Cantor set $C_a$ for $a=\frac{3}{10}$.
  • Figure 3: The triangle $Talpha,lambda$ with vertices $x,y,z$, largest angle at $z$, height $\alpha$, and base $1$.
  • Figure 4: Parent square $S_\emptyset$ and first-generation children of radius $\rho$ for self-similar compact set $C$ in the infinity norm $\|\cdot\|_\infty$.
  • Figure 5: Parent square $S_\emptyset$ and first-generation children of radius $\rho$ for randomly perturbed self-similar compact set $C$ in the infinity norm $\|\cdot\|_\infty$.
  • ...and 3 more figures

Theorems & Definitions (48)

  • Definition 1.1
  • Proposition 1.2: Yavicoli Yavicoli_Survey
  • Lemma 1.3
  • Proposition 2.1: Convex combinations in $\mathbb{R}$
  • Theorem 2.2: 3-point configurations in $C\times C$
  • Corollary 2.2.1
  • Remark 2.3: Our result is a first of its kind in $\mathbb{R}^2$
  • Remark 2.4: Comparison between Newhouse thickness and Hausdorff dimension
  • Remark 2.5: Longer progressions
  • Remark 2.6: Sharpness of our result: the off-center Cantor set has thickness 1 and no $4$-AP
  • ...and 38 more