On the Hausdorff dimension of the residual set of a packing by smooth curves

TL;DR

This paper investigates the Hausdorff dimension of residual sets in planar packings by smooth curves with curvature bounds, establishing lower bounds away from 1 and demonstrating optimality through counterexamples.

Contribution

It proves that packings by curves with a lower curvature bound have residual sets with Hausdorff dimension bounded away from 1, extending previous results to more general packings.

Findings

Residual sets of packings with curvature bounds have dimension > 1

Counterexamples show dimension can be 1 without curvature bounds

Packings by convex curves cannot have σ-finite Hausdorff 1-measure

Abstract

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $\sigma$-finite Hausdorff 1-measure.