Structure of sets with nearly maximal Favard length

Abstract

Let E ⊂ B(1) ⊂ ‫ޒ‬ 2 be an H 1 measurable set with H 1 (E) < ∞, and let L ⊂ ‫ޒ‬ 2 be a line segment with H 1 (L) = H 1 (E). It is not hard to see that Fav(E) ≤ Fav(L). We prove that in the case of near equality, that is,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 = Cδ 1/70 for an absolute constant C > 0.