On k-point configuration sets with nonempty interior

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 Sjolin (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, E1,MIDLINE HORIZONTAL ELLIPSIS,Ek$E_1,\,\dots ,\, E_k$, 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 R2$\mathbb {R}<^>2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in R3$\mathbb {R}<^>3$; and ratios of pinned distances in R2$\mathbb {R}<^>2$ and R3$\mathbb {R}<^>3$. Results for 4-point configurations include cross-ratios on R$\mathbb {R}$, pairs of areas of triangles determined by quadrilaterals in R2$\mathbb {R}<^>2$, and dot products of differences in Rd$\mathbb {R}<^>d$.