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Falconer-type results for any finite graph with multiple pins

Tainara Borges, Ben Foster, Yumeng Ou, Eyvindur Palsson, Francisco Romero Acosta

Abstract

A generalization of the celebrated Falconer distance problem asks for a graph $G=(\mathcal{V},\mathcal{E})$, with vertex set $\mathcal{V}$ and edge set $\mathcal{E}$, how large the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$, $d\geq 2$, needs to be to guarantee that the distance graph $$ Δ^{G}(E):= \lbrace (|x_{i}-x_{j}|)_{(v_i,v_j)\in\mathcal{E}} : x_1,\ldots,x_{|\mathcal{V}|}\in E \rbrace $$ has positive $|\mathcal{E}|$-dimensional Lebesgue measure. Here we represent the edges in $\mathcal{E}$ as ordered pairs of vertices $(v_i,v_j)$ with $i<j$. Many results exist for particular graphs, such as trees and simplices. Some general results exist, but they require intricate calculations, such as computing Fourier decay of the natural measure on the configuration set or mapping properties of associated Fourier integral operators. In this paper, using the graph theory notion of $k$-degeneracy, which is easy to compute, we obtain a non-trivial dimensional threshold $\frac{d+k}{2}$, $d>k$, for any non-trivial graph $G$. Key ingredients for our result are identifying pinned stars as the right building blocks for a general graph as well as refining a Fubini type argument due to Taylor and the third named author. We further generalize this to graphs with multiple pins by introducing the $k$-admissibility of a graph, a generalization of $k$-degeneracy that takes pins into account, as well as by extending the Fubini argument to the multiple pinned setting. Not only do we obtain non-trivial results in high enough dimensions for any distance graph, but for particular graphs (such as cycles) our results are also strong and improve the previously best known results. Our methods extend to general two point configurations, contingent on results being available for the appropriate star building blocks.

Falconer-type results for any finite graph with multiple pins

Abstract

A generalization of the celebrated Falconer distance problem asks for a graph , with vertex set and edge set , how large the Hausdorff dimension of a compact set , , needs to be to guarantee that the distance graph has positive -dimensional Lebesgue measure. Here we represent the edges in as ordered pairs of vertices with . Many results exist for particular graphs, such as trees and simplices. Some general results exist, but they require intricate calculations, such as computing Fourier decay of the natural measure on the configuration set or mapping properties of associated Fourier integral operators. In this paper, using the graph theory notion of -degeneracy, which is easy to compute, we obtain a non-trivial dimensional threshold , , for any non-trivial graph . Key ingredients for our result are identifying pinned stars as the right building blocks for a general graph as well as refining a Fubini type argument due to Taylor and the third named author. We further generalize this to graphs with multiple pins by introducing the -admissibility of a graph, a generalization of -degeneracy that takes pins into account, as well as by extending the Fubini argument to the multiple pinned setting. Not only do we obtain non-trivial results in high enough dimensions for any distance graph, but for particular graphs (such as cycles) our results are also strong and improve the previously best known results. Our methods extend to general two point configurations, contingent on results being available for the appropriate star building blocks.
Paper Structure (14 sections, 15 theorems, 49 equations, 15 figures)

This paper contains 14 sections, 15 theorems, 49 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Notion of admissibility and some key examples
  4. Allowed moves: definition of $k$-admissible
  5. Admissibility for some key examples
  6. Complete graph in $(k+1)$ vertices
  7. Chains
  8. Trees
  9. Multiply-pinned cycles
  10. Trees of triangles
  11. Double banana
  12. An algorithm - graph dismantling lemma
  13. Special case: Multiple pinned chains and cycles
  14. Proof of general structural Theorems \ref{['thm: firststructuralthm']} and \ref{['thm: secondstructuralthm']}

Key Result

Theorem 2

Let $G=(\mathcal{V},\mathcal{E})$ be a non-trivial graph that is $k$-degenerate. Then for any compact set $E\subset\mathbb{R}^d$, $d>k$, with $\dim(E)>\frac{d+k}{2}$ the $|\mathcal{E}|$-dimensional Lebesgue measure of $\Delta^{G}(E)$ is positive.

Figures (15)

  • Figure 1: $S_7$, the $7$-star graph, with pins at each leaf.
  • Figure 2: Pinned triangle.
  • Figure 3: Pinned complete graph in $5$ vertices.
  • Figure 4: Example of multiply pinned chain.
  • Figure 5: Pinned tree with $5$ pins and admissibility number $3$.
  • ...and 10 more figures

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Definition 4
  • Definition 5
  • Theorem 6: IPPS22
  • Remark 7
  • Theorem A: Structural theorem for Euclidean distances
  • Corollary 8: Corollary of the structural theorem for cycles
  • Definition 9
  • ...and 30 more