  The Besicovitch projection theorem states that if a subset $E$ of the plane
has finite length in the sense of Hausdorff measure and is purely unrectifiable
(so its intersection with any Lipschitz graph has zero length), then almost
every orthogonal projection of $E$ to a line will have zero measure. In other
words, the Favard length of a purely unrectifiable $1$-set vanishes. In this
article, we show that when linear projections are replaced by certain nonlinear
projections called curve projections, this result remains true. In fact, we go
further and use multiscale analysis to prove a quantitative version of this
Besicovitch nonlinear projection theorem. Roughly speaking, we show that if a
subset of the plane has finite length in the sense of Hausdorff and is nearly
purely unrectifiable, then its Favard curve length is very small. Our
techniques build on those of Tao, who in [Tao09] proves a quantification of the
original Besicovitch projection theorem.
