English

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 ϵ>0\epsilon > 0 the proof is able to give an explicit description of the tubes for which the total width is less than ϵ.\epsilon.

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

R2 v1 2026-06-28T20:56:26.635Z