Quantifying the Besicovitch projection theorem

Abstract

The Besicovitch projection theorem, which is one of the cornerstones of geometric measure theory, states the following: a set $E$ of finite length is purely unrectifiable if and only if almost every orthogonal projection of $E$ is Lebesgue-null. This old and remarkable result, which links rectifiability and projections, is purely qualitative - it does not provide any bounds for the size of projections of sets that are ``close to being purely unrectifiable.'' In the last 30 years significant effort has been put into quantifying Besicovitch’s theorem, and the main motivation is the Vitushkin’s conjecture on removable sets for bounded analytic functions. In this mini-course I will present the history of this problem, as well as some tools and ideas involved in the most recent progress.

Damian Dąbrowski

Assistant professor

My research interests include geometric measure theory and harmonic analysis.