Given a subset Y of the plane and a point x not in Y, we can consider the visibility of Y from the point x, which is the size of the radial projection of Y onto the unit circle centered at x. If Y is a large set, then it is possible for the visibility of Y to be small at a few points, but it is difficult for the visibility to be small at many points. To make this intuition precise, we run into a variant of Bourgain's discretized sum-product theorem. I will discuss some (limited) progress on the visibility problem, as well as some hopes on how to move forward.
Back to Workshop III: The Kakeya Problem, Restriction Problem, and Sum-product Theory