On $k$-point configuration sets with nonempty interior

Abstract

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sjölin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\dots ,E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in ${\mathbb{R}}^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in ${\mathbb{R}}^3$; and ratios of pinned distances in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$. Results for 4-point configurations include cross-ratios on $\mathbb{R}$, triangle area pairs determined by quadrilaterals in ${\mathbb{R}}^2$, and dot products of differences in ${\mathbb{R}}^d$.

31 pages, no figures, minor revision for publication

References in corpus (2)

• Configuration sets with nonempty interior
• Existence of similar point configurations in thin subsets of ${\mathbb{R}}^d$

Cited by in corpus (1)

• Infinite constant gap length trees in products of thick Cantor sets