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 a planar set E is AD-regular and there exists a set of direction G with ${\mathcal{H}}^1(G)\gtrsim 1$ such that for every $\theta \in G$ we have $||\pi \theta {\mathcal{H}}^1|E||L^{\mathrm{\infty }}\lesssim 1$, then a big piece of E can be covered by a Lipschitz graph $\mathrm{\Gamma }$ with $Lip(\mathrm{\Gamma })\lesssim 1$. 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.