
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 the proof is able to give an explicit description of the tubes for which the total width is less than
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Taxonomy
TopicsMarine Sponges and Natural Products
