Super-uniformity of the typical billiard path (proof included)

Video of this lecture

COMPUTER SCIENCE AND DISCRETE MATHEMATICS I
Topic:Super-uniformity of the typical billiard path (proof included)
Speaker:Jozsef Beck
Affiliation:Rutgers, The State University of New Jersey
Date:Monday, October 4
Time/Room:11:15am - 12:15pm/S-101
Video Link:https://video.ias.edu/csdm/beck

I will describe the proof of the following surprising result: the typical billiard paths form the family of the most uniformly distributed curves in the unit square. I will justify this vague claim with a precise statement. As a byproduct, we obtain the counter-intuitive fact that the complexity of the test set is almost irrelevant. The error term is shockingly small, and it does not matter that we test uniformity with a nice set (like a circle or a square), or with an arbitrarily ugly Lebesgue measurable subset of the unit square.