Covering the Sierpi\'nski carpet 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.