Curve packing and modulus estimates

TL;DR

This paper investigates the geometric properties of Moser families of planar curves, proving that their p-modulus is bounded below by a positive constant for p > 3, strengthening previous area-based results.

Contribution

It extends Marstrand's classical result by establishing a positive lower bound on the p-modulus of Moser families for p > 3, providing a new quantitative measure of their size.

Findings

p-modulus of Moser families is at least c_p > 0 for p > 3

The union of curves in a Moser family has non-zero area

Strengthens classical area lower bounds with modulus estimates

Abstract

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $\mathbb{R}^{2}$ of length one. The classical "worm problem" of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family has always area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand's result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$.