Covering the Sierpiński carpet with tubes
Abstract
We show that nontrivial $\times N$invariant sets in $[0,1]^d$, such as the Sierpiński carpet and the Sierpiński sponge, are tubenull, that is, they can be covered by a union of tubular neighbourhoods of lines of arbitrarily small total volume. This introduces a new class of tubenull sets of dimension strictly between $d1$ and $d$. We utilize ergodictheoretic methods to decompose the set into finitely many parts, each of which projects onto a set of Hausdorff dimension less than $1$ in some direction. We also discuss coverings by tubes for other selfsimilar sets, and present various applications.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2006.00499
 arXiv:
 arXiv:2006.00499
 Bibcode:
 2020arXiv200600499P
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Dynamical Systems;
 Mathematics  Metric Geometry;
 Primary 37C45;
 Secondary 28A80
 EPrint:
 24 pages, 2 figures