2020/05/31 by Aleksi Pyörälä, Pablo Shmerkin, Pyörälä, Aleksi +5
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary 37C45 #Secondary 28A80 #Topological and Geometric Data Analysis #math.CA #math.DS #math.MG #msc:28A80 #msc:37C45
paper · pdf · doi:10.48550/arxiv.2006.00499
24 pages, 2 figures
arxiv created 2020/05/31 · openalex publication_date 2020/05/31 · arxiv updated 2020/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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. 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.