Speaker
Joshua Zahl, Nankai University
Abstract
Nikodym constructed a measure-zero Borel set in the plane that contains a punctured line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when “lines through a point” are replaced by ”circles or circular arcs with a prescribed center.” This was answered in the negative by Bourgain and Marstrand. I will discuss a generalization of this question, in which “lines through a point” or “circles with prescribed center” are replaced by more general analytic families of plane curves.