  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
$\|\pi_\theta\mathcal{H}^1|_E\|_{L^{\infty}}\lesssim 1$, 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.
