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 ${\mathbb{R}}^d$ 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 $d\ge 2$; 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.