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

TL;DR

This paper establishes explicit conditions under which thick, compact subsets of Euclidean spaces contain specific 3-point configurations, including arithmetic progressions and similar triangles, advancing understanding of geometric patterns in fractal-like sets.

Contribution

It provides the first explicit criteria for the presence of 3-point configurations in planar sets based on thickness and uniformity conditions.

Findings

Sets with sufficient thickness contain similar copies of any 3-point configuration.

Compact sets in the plane contain vertices of any given triangle if they meet certain thickness criteria.

Product sets like C×C contain 3-point configurations if the thickness exceeds a threshold.

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.