Rigidity, graphs and Hausdorff dimension

TL;DR

This paper investigates the relationship between the Hausdorff dimension of a set and the richness of geometric configurations defined by graphs within that set, establishing conditions for positive measure of these configurations.

Contribution

It introduces a framework linking graph-based point configurations to Hausdorff dimension, proving a threshold condition for the abundance of such configurations in fractal sets.

Findings

Existence of a dimension threshold $s_k<d$ for positive measure of configurations.

Embedding of configuration equivalence classes into Euclidean space based on graph edges.

Use of combinatorial, topological, and analytic methods to establish results.

Abstract

For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding vertices of $G$ are connected by an edge. We regard two such frameworks as equivalent if the specified distances are the same. We show that in a suitable sense the set of equivalences of such frameworks naturally embeds in ${\mathbb R}^m$ where $m$ is the number of "essential" edges of $G$. We prove that there exists a threshold $s_k<d$ such that if the Hausdorff dimension of $E$ is greater than $s_k$, then the $m$-dimensional Hausdorff measure of the set of equivalences of $G$-frameworks is positive. The proof relies on combinatorial, topological and analytic considerations.