UBC Math Department Colloquium: Krystal Taylor

Measure zero sets containing intricate structure are foundational in geometric measure theory. These include Besicovitch sets in the plane (also known as Kakeya sets), which are measure zero sets containing a line in every direction. Closely related to the concept of Besicovitch sets is Davies' efficient covering theorem, which states that an arbitrary measurable set A in the plane can be covered by full lines so that the union of the lines has the same measure as A. This result has an interesting dual formulation in the form of a prescribed projection theorem. It turns out that Kakeya sets can be constructed using a prescribed projection theorem. We take a deeper look at these connections and consider relevant tools and examples along the way.