Covering sponges with tubes
Classical Analysis and ODEs
2025-11-24 v2 Dynamical Systems
Metric Geometry
Abstract
The aim of this note is to give a short proof of a result of Py\"or\"al\"a--Shmerkin--Suomala--Wu; the Sierpi\'nski carpet, and generalisations, are tube-null; they can be covered with tubes of arbitrarily small total width. We remark that a more general class of sponge-like sets satisfy this property. For a given the proof is able to give an explicit description of the tubes for which the total width is less than
Cite
@article{arxiv.2501.02374,
title = {Covering sponges with tubes},
author = {William O'Regan},
journal= {arXiv preprint arXiv:2501.02374},
year = {2025}
}
Comments
13 pages, accepted version