Covering sponges with tubes

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 $\epsilon.$