Abstract:

The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a line dropped randomly onto the plane intersects such a set E is equal to zero. Thus, the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. We discuss generalized Besicovitch theorems and, as Tao did for the classical theorem (Proc. London Math. Soc., 2009), we use multi-scale analysis to quantify these results. This work is joint with Laura Cladek and Krystal Taylor.