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 ${\mathcal{H}}^1(E)<\infty$, 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, $$\mathrm{Fav}(E)\ge \mathrm{Fav}(L)-\hat{I}´,$$ the set $E$ can be covered by an $\hat{I}\mu$-Lipschitz graph, up to a set of length $\hat{I}\mu$. The dependence between $\hat{I}\mu$ and $\hat{I}´$ is polynomial: in fact, the conclusions hold with $\hat{I}\mu =C\hat{I}{´}^{1/70}$ for an absolute constant $C>0$.

25 pages, 4 figures