Structure of sets with nearly maximal Favard length

Abstract

Let $E\subset B(1)\subset {\mathbb{R}}^2$ be an ${\mathcal{H}}^1$ measurable set with ${\mathcal{H}}^1(E)<\mathrm{\infty }$, and let $L\subset {\mathbb{R}}^2$ be a line segment with ${\mathcal{H}}^1(L)={\mathcal{H}}^1(E)$. It is not hard to see that $\mathrm{Fav}(E)\le \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$\mathrm{Fav}(E)\ge \mathrm{Fav}(L)-\delta ,$$ the set $E$ can be covered by an $ϵ$-Lipschitz graph, up to a set of length $ϵ$. The dependence between $ϵ$ and $\delta$ is polynomial: in fact, the conclusions hold with $ϵ=C{\delta }^{1/70}$ for an absolute constant $C>0$.