The Favard length of a planar Borel set is the average length of its orthogonal projections. We prove that if an Ahlfors 1-regular set has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of …
Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.
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 …