Favard length of random Cantor sets

Abstract

Favard length of a planar compact set is the average length of its orthogonal projections. It has been known for a long time that given a nice enough self-similar set of dimension 1, such as the classical 4-corners Cantor set, its Favard length is zero. Consequently, the Favard length of delta-neighbourhoods of such sets converges to 0 as delta goes to 0. The ‘Favard length problem’, first posed by Peres and Solomyak in 2002, asks about the rate of convergence. The original motivation comes from the Vitushkin’s conjecture in complex analysis. This talk will be a gentle introduction to this topic. Time permitting, I will also report on an ongoing work with Alan Chang and Giacomo Del Nin on some random variant of the problem.

Damian Dąbrowski

Assistant professor

My research interests include geometric measure theory and harmonic analysis.