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 that A satisfies a mild Yavicoli-thickness condition and an r -uniformity condition for $d\geq 2$ ; or, when $d=1$ , the result holds provided that 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.