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

arXiv comment25 pages, 4 figures

References in corpus (5)

• A quantitative version of the Besicovitch projection theorem via multiscale analysis
• Characterising the big pieces of Lipschitz graphs property using projections
• The Favard length of product Cantor sets
• Plenty of big projections imply big pieces of Lipschitz graphs
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Cited by in corpus (2)

• Quantitative Besicovitch projection theorem for irregular sets of directions
• Analytic capacity and dimension of sets with plenty of big projections