Covering sponges with tubes

TL;DR

This paper provides a concise proof that the Sierpiński carpet and similar sponge-like fractals can be covered by tubes of arbitrarily small total width, extending previous results and offering explicit constructions.

Contribution

It offers a simplified proof that sponge-like sets are tube-null and generalizes the result to a broader class of fractals, with explicit tube constructions.

Findings

Sierpiński carpet is tube-null.

General sponge-like sets are also tube-null.

Explicit tube coverings with arbitrarily small total width are constructed.

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