Falconer-type results for any finite graph with multiple pins

TL;DR

This paper extends Falconer-type results to arbitrary finite graphs with multiple pins, establishing new Hausdorff dimension thresholds for the positivity of the associated distance graph measures, using graph degeneracy and refined analytical techniques.

Contribution

It introduces the concepts of $k$-admissibility and pin-based analysis to derive dimension thresholds for any finite graph, improving upon previous results and simplifying calculations.

Findings

Established a dimension threshold of (d+k)/2 for non-trivial graphs using $k$-degeneracy.

Extended results to graphs with multiple pins via $k$-admissibility.

Improved bounds for specific graphs like cycles.

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 $$ \Delta^{G}(E):= \lbrace (|x_{i}-x_{j}|)_{(v_i,v_j)\in\mathcal{E}} : x_1,\ldots,x_{|\mathcal{V}|}\in E \rbrace $$ 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.