Covering the Sierpiński carpet with tubes

We show that non-trivial ×N-invariant sets in [0, 1]^d, such as the Sierpiński carpet and the Sierpiński 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, and continues the study of the geometric properties of dynamically defined sets. 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.