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.

arXiv comment18 pages, 2 figures; discussion of sharpness addded

References in corpus (5)

• Weighted restriction estimates and application to Falconer distance set problem
• On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators
• An elementary approach to simplexes in thin subsets of Euclidean space
• Simplices in thin subsets of Euclidean spaces
• Dimensional lower bounds for Falconer type incidence and point configuration theorems