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Covering sponges with tubes

William O'Regan

Abstract

The aim of this note is to give a short proof of a result of Pyörälä--Shmerkin--Suomala--Wu; the Sierpiński carpet, and generalisations, are tube-null; they can be covered with tubes of arbitrarily small total width. We remark that a more general class of sponge-like sets satisfy this property. For a given $ε> 0$ the proof is able to give an explicit description of the tubes for which the total width is less than $ε.$

Covering sponges with tubes

Abstract

The aim of this note is to give a short proof of a result of Pyörälä--Shmerkin--Suomala--Wu; the Sierpiński carpet, and generalisations, are tube-null; they can be covered with tubes of arbitrarily small total width. We remark that a more general class of sponge-like sets satisfy this property. For a given the proof is able to give an explicit description of the tubes for which the total width is less than
Paper Structure (10 sections, 20 theorems, 64 equations)

This paper contains 10 sections, 20 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The localisation problem
  3. The definition of a tube-null set
  4. Examples of tube-null sets
  5. Preliminaries and notation
  6. Sketch of the proof
  7. Projections of $\times N$-invariant measures
  8. The Sierpiński carpet is tube-null
  9. Final remarks
  10. Other tube-null sets

Key Result

Proposition 1.5

Let $K \subset \mathbb{R}^d.$ Suppose there exists a countable decomposition a countable family of $d-1$-dimensional hyperplanes $\{V_n\}_{n \in \mathbb{N}},$$V_n \in G(d,d-1),$ and orthogonal projections $P_{V_n}:\mathbb{R}^d \rightarrow V_n$ with $\mathcal{L}^{d-1}(P_{V_n}(K_n)) = 0.$ Then $K$ is tube-null.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • Theorem 1.7: Theorem 1.1, har
  • Theorem 1.9: Theorem 1.1, pyo
  • Definition 1.10
  • ...and 33 more