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arXiv · 2006.00499

Covering the Sierpi\'nski carpet with tubes

Aleksi Pyörälä·Pablo Shmerkin·Ville Suomala·Meng Wu
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Abstract

We show that non-trivial $\times N$-invariant sets in $[0,1]^d$, such as the Sierpi\'{n}ski carpet and the Sierpi\'{n}ski sponge, are tube-null, 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 tube-null sets of dimension strictly between $d-1$ and $d$. We utilize ergodic-theoretic 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 self-similar sets, and present various applications.

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Aleksi Pyörälä, Pablo Shmerkin, Ville Suomala, Meng Wu. 2020-05-31. Covering the Sierpi\'nski carpet with tubes. https://arxiv.org/abs/2006.00499

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