Events for 05/02/2024 from all calendars

Working Seminar on Banach and Metric Spaces

Time: 10:00AM - 11:30PM

Location: BLOC 302

Speaker: Cosmas Kravaris, Princeton University

Title: Lower bounds on the discrepancy of axis parallel rectangles on the plane

Abstract: In 1954, Roth proved that for any N points on the unit square [0,1]^2, there is a rectangle [0,q_1]x[0,q_2] inside [0,1]^2 such that the number of points it contains differs from q_1 q_2 N by at least the Ω(sqrt(log N)). Hence there is a limit to how well a finite set of points on the unit square can approximate the uniform measure. The bound was improved by Schmidt in 1972 to Ω(log N), which is known to be sharp. In this talk, we will prove these two results, by exploiting the structure of the Haar basis on the plane. The bound of Roth states that if we pick an axis parallel rectangle at random, then the discrepancy squared is bounded from below (e.g. an L^2 norm estimate on the discrepancy). If time permits, we discuss how this result generalizes to L^p moments using the Littlewood-Paley inequalities.