Local entropy and projections of dynamically defined fractals

Kevin Hughes, Princeton University
Fine Hall 314

What's the smallest area a subset of the plane containing a unit line segment in every direction can have? Besicovitch showed that you can get 0 measure and Fefferman used the existence of this set to provide a solution to the ball multiplier problem. Kakeya sets have been important in understanding other phenomena like the Bochner-Riesz conjecture as the Hausdorff/Minkowski dimension of Kakeya sets relate to the two. Building on Fefferman's work, Bourgain improved our understanding introducing an ingenious bush argument . I will use the analytic story to motivate the corresponding problems for varieties over finite fields and recent work done there.