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 Sj\"olin \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, E1β,β¦,Ekβ, 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
R2 or the radii of their circumscribing circles; volumes of pinned
parallelepipeds in R3; and ratios of pinned distances in R2 and R3. Results for 4-point configurations include cross-ratios
on R, triangle area pairs determined by quadrilaterals in R2, and dot products of differences in Rd.Comment: 32 pages, no figure