A quantification of the Besicovitch projection theorem and its generalizations (part 2).
Talk by Dr. Blair Davey (Mathematical Sciences Dept., MSU)
9/21/2020 4:10-5:00pm Webex Mtg.
Abstract: (2nd Part of a 2-Part Talk) - 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 randomly dropped line intersects such a set E is equal to zero. This shows us that 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. This leads us to discuss generalized Besicovitch theorems and the ways in which we can quantify such results by building upon the work of Tao, Volberg, and others. This talk covers joint work with Laura Cladek and Krystal Taylor.