Nonempty interior of configuration sets via microlocal partition optimization

We give sufficient Hausdorff dimensional conditions for a $k$-point configuration set generated by elements of thin sets in $\mathbb R^d$ to have nonempty interior. In earlier work \cite{GIT19,GIT20}, we extended Mattila and Sj\"olin's theorem concerning distance sets in Euclidean spaces \cite{MS99} to $k$-point configurations in general manifolds. The dimensional thresholds in \cite{GIT20} were dictated by associating to a configuration function a family of generalized Radon transforms and then optimizing $L^2$-Sobolev estimates for them over all nontrivial bipartite partitions of the $k$ points. In the current work, we extend this by allowing the optimization to be carried out locally over the configuration's incidence relation, or even microlocally over the conormal bundle of the incidence relation. To illustrate this approach, we apply it to (i) areas of subtriangles determined by quadrilaterals and pentagons in a set $E\subset\mathbb R^2$; (ii) pairs of ratios of pinned distances in $\mathbb R^d$; and (iii) a short proof of Palsson and Romero Acosta's result \cite{PRA21} on congruence classes of triangles in $\mathbb R^d$.