On <i>k</i> ‐point configuration sets with nonempty interior
Allan Greenleaf, Alex Iosevich, Krystal Taylor
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 (J. Geom. Anal. 31 (2021), 6662–6680) on 2-point configurations, extending a theorem of Mattila and Sjölin (Math. Nachr. 204 (1999), 157–162) 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 , ⋯ , E k $E_1,\,\dots ,\, E_k$ , has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing L2-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 R 2 $\mathbb {R}^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in R 3 $\mathbb {R}^3$ ; and ratios of pinned distances in R 2 $\mathbb {R}^2$ and R 3 $\mathbb {R}^3$ . Results for 4-point configurations include cross-ratios on R $\mathbb {R}$ , pairs of areas of triangles determined by quadrilaterals in R 2 $\mathbb {R}^2$ , and dot products of differences in R d $\mathbb {R}^d$ .