Quantitative Besicovitch projection theorem for irregular sets of directions

Abstract The classical Besicovitch projection theorem states that if a planar set with finite length is purely unrectifiable, then almost all orthogonal projections of have zero length. We prove a quantitative version of this result: if is AD‐regular and there exists a set of direction with such that for every we have , then a big piece of can be covered by a Lipschitz graph with . The main novelty of our result is that the set of good directions is assumed to be merely measurable and large in measure, while previous results of this kind required 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 is contained in some Lipschitz graph if and only if the complement of the set of directions spanned by contains an arc.