  By using a multiscale analysis, we establish quantitative versions of the
Besicovitch projection theorem (almost every projection of a purely
unrectifiable set in the plane of finite length has measure zero) and a
standard companion result, namely that any planar set with at least two
projections of measure zero is purely unrectifiable. We illustrate these
results by providing an explicit (but weak) upper bound on the average
projection of the $n^{th}$ generation of a product Cantor set.
