Quantitative Besicovitch projection theorem for irregular sets of directions

Abstract

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if $E\subset {\mathbb{R}}^2$ is AD-regular and there exists a set of direction $G\subset {\mathbb{S}}^1$ with ${\mathcal{H}}^1(G)\gtrsim 1$ such that for every $\theta \in G$ we have $\Vert {\pi }_{\theta }{\mathcal{H}}^1{|}_E{\Vert }_{L^{\infty }}\lesssim 1$, then a big piece of $E$ can be covered by a Lipschitz graph $\Gamma$ with $\mathrm{Lip}(\Gamma )\lesssim 1$. The main novelty of our result is that the set of good directions $G$ is assumed to be merely measurable and large in measure, while previous results of this kind required $G$ to be an arc. As a corollary, we obtain a result on AD-regular sets which avoid a large set of directions, in the sense that the set of directions they span has a large complement. It generalizes the following easy observation: a set $E$ is contained in some Lipschitz graph if and only if the complement of the set of directions spanned by $E$ contains an arc.

arXiv commentv2: minor revision, accepted version, 52 pages

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