Covering the Sierpi\'nski carpet with tubes

TL;DR

This paper proves that certain fractal sets like the Sierpiński carpet can be covered by tubes of arbitrarily small total volume, introducing a new class of such sets with intermediate dimension.

Contribution

It establishes that non-trivial N-invariant fractals are tube-null, expanding the understanding of their geometric measure properties using ergodic theory.

Findings

Sierpinski carpet is tube-null.

Introduces a new class of tube-null sets between dimensions d-1 and d.

Provides methods for covering self-similar sets with tubes.

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.