Covering sponges with tubes

Abstract

The aim of this note is to give a short proof of a result of Pyörälä–Shmerkin–Suomala–Wu; the Sierpiński 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 \(\epsilon\).