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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

Let E B(1) 2 be an 1 measurable set with 1(E) < , and let L 2 be a line segment with 1(L) = 1(E). It is not hard to see that Fav (E) Fav (L). We prove that in the case of near equality, that is,

Fav (E) 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 = Cδ170 for an absolute constant C > 0.

Favard length, Besicovitch projection theorem, Lipschitz graph
Mathematical Subject Classification
Primary: 28A75
Secondary: 28A78
Received: 7 April 2022
Revised: 6 August 2022
Accepted: 27 October 2022
Published: 17 May 2024
Alan Chang
Department of Mathematics
Washington University in St. Louis
United States
Damian Dąbrowski
Department of Mathematics and Statistics
University of Jyväskylä
Tuomas Orponen
Department of Mathematics and Statistics
University of Jyväskylä
Michele Villa
Research Unit of Mathematical Sciences
University of Oulu

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