A Mattila-Sjölin theorem for triangles

Abstract

We show for a compact set $E\subset {\mathbb{R}}^d$, $d\ge 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of $E$ as the set $${\Delta }_{\text{tri}}(E)=\left\{(t,r,\alpha ):|x-z|=t,|y-z|=r\text{ and }\alpha =\alpha (x,z,y),x,y,z\in E\right\},$$ where $\alpha (x,z,y)$ denotes the angle formed by $x$, $y$ and $z$ , centered at $z$. This extends the Mattila-Sjölin theorem that establishes a non-empty interior for the distance set instead of the set of congruence classes of triangles. These theorems can be thought of as refinements and extensions of the statements in the well known Falconer distance problem.

18 pages, 2 figures; discussion of sharpness addded

References in corpus (2)

• An elementary approach to simplexes in thin subsets of Euclidean space
• Simplices in thin subsets of Euclidean spaces