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Nonempty interior of configuration sets via microlocal partition optimization

Allan Greenleaf, Alex Iosevich, Krystal Taylor

Abstract

We prove new results of Mattila-Sjölin 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-Sjölin 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 $\Bbb R^d$.

Nonempty interior of configuration sets via microlocal partition optimization

Abstract

We prove new results of Mattila-Sjölin type, giving lower bounds on Hausdorff dimensions of thin sets ensuring that various -point configuration sets, generated by elements of , 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 -Sobolev estimates for them over all nontrivial bipartite partitions of the 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-Sjölin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set ; (ii) pairs of ratios of distances of 4-tuples in ; and (iii) similarity classes of triangles in , as well as to (iv) give a short proof of Palsson and Romero Acosta's result on congruence classes of triangles in .
Paper Structure (9 sections, 9 theorems, 94 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $k$-point $\Phi$-configuration sets
  3. Areas of triangles
  4. Pairs of areas of triangle in quadrilaterals
  5. Triples of areas of triangles in quadrilaterals
  6. Triples of areas of triangles in pentagons
  7. Pairs of ratios of distances
  8. Congruence classes of triangles in $\Bbb R^d$
  9. Similarity classes of triangles in $\Bbb R^d$

Key Result

Theorem 1.1

If $E\subset\Bbb R^2$ is compact, then (i) if $\hbox{dim}_{\mathcal{H}}(E)>3/2$, then $\hbox{Int}\,\left\{(|xyz|,|xzw|)\in\Bbb R^2: x,y,z,w\in E\right\}\ne\emptyset$; (ii) if $\hbox{dim}_{\mathcal{H}}(E)>7/4$, then $\hbox{Int}\,\left\{(|xyz|,|xzw|,|xyw|)\in\Bbb R^3: x,y,z,w\in E\right\}\ne\emptyset$

Figures (1)

  • Figure 1: Two pairs of ratios of distances of points in a 4-tuple. Figure on left corresponds to \ref{['thm 1.2 eqn']} from Thm. \ref{['thm ratios']}, with ratio of the two blue side lengths forming the first coordinate of $\Phi(x,y,z,w)$ and the ratio of the two red side lengths being the second coordinate. Figure on right illustrates \ref{['eqn zywy']} for the same 4 points, with the two coordinates of $\Phi(x,y,z,w)$ being the ratios of the lengths of the blue segments to the dotted one.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 5 more