Structure of sets with nearly maximal Favard length

Chang, Alan; Dąbrowski, Damian; Orponen, Tuomas; Villa, Michele

doi:10.2140/apde.2024.17.1473

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Structure of sets with nearly maximal Favard length

Alan Chang, Damian Dąbrowski, Tuomas Orponen and Michele Villa

Vol. 17 (2024), No. 4, 1473–1500

DOI: 10.2140/apde.2024.17.1473

Abstract

Let $E \subset B (1) \subset ℝ^{2}$ be an $ℋ^{1}$ measurable set with $ℋ^{1} (E) < \infty$ , and let $L \subset ℝ^{2}$ be a line segment with $ℋ^{1} (L) = ℋ^{1} (E)$ . It is not hard to see that $Fav (E) \leq Fav (L)$ . We prove that in the case of near equality, that is,

Fav (E) \geq Fav (L) - δ,

the set $E$ can be covered by an $𝜖$ -Lipschitz graph, up to a set of length $𝜖$ . The dependence between $𝜖$ and $δ$ is polynomial: in fact, the conclusions hold with $𝜖 = C δ^{1 ∕ 70}$ for an absolute constant $C > 0$ .

Keywords

Favard length, Besicovitch projection theorem, Lipschitz graph

Mathematical Subject Classification

Primary: 28A75

Secondary: 28A78

Milestones

Received: 7 April 2022

Revised: 6 August 2022

Accepted: 27 October 2022

Published: 17 May 2024

Authors

Alan Chang
Department of Mathematics Washington University in St. Louis MO United States

Damian Dąbrowski
Department of Mathematics and Statistics University of Jyväskylä Finland

Tuomas Orponen
Department of Mathematics and Statistics University of Jyväskylä Finland

Michele Villa
Research Unit of Mathematical Sciences University of Oulu Finland

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Vol. 17, No. 4, 2024