Nonempty interior of configuration sets via microlocal partition optimization

TL;DR

This paper develops a microlocal optimization approach to establish lower bounds on the Hausdorff dimensions of sets ensuring certain geometric configurations have nonempty interior, extending previous Radon transform methods.

Contribution

It introduces a microlocal partition optimization technique for configuration sets, improving dimension estimates for geometric configurations in Euclidean spaces.

Findings

Proved nonempty interior results for areas of subtriangles in sets in R^2.

Established dimension thresholds for ratios of distances in 4-tuples in R^d.

Extended results to similarity classes of triangles and provided a new proof for congruence classes.

Abstract

We prove new results of Mattila-Sj\"olin type, giving lower bounds on Hausdorff dimensions of thin sets $E\subset \Bbb R^d$ ensuring that various $k$-point configuration sets, generated by elements of $E$, have nonempty interior. The dimensional thresholds in our previous work \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 done locally over the configuration's incidence relation, or even microlocally over the conormal bundle of the incidence relation. We use this approach to prove Mattila-Sj\"olin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set $E\subset\Bbb R^2$; (ii) pairs of ratios of distances of 4-tuples in $\Bbb R^d$; and (iii) similarity classes of triangles in $\Bbb R^d$, as well as to (iv) give a short proof of Palsson and Romero Acosta's result on congruence classes of triangles in $\Bbb R^d$.