Recent advances in geometric discrepancy

Abstract: 

Classical discrepancy theory deals with distributing NN points in the ss-dimensional unit cube as evenly as possible. The discrepancy function of the point set measures the deviation of the empirical measure from the uniform measure with respect to rectangular boxes anchored at the origin, and the LpLp discrepancy is the LpLp norm of the discrepancy function. While explicit constructions of point sets with asymptotically optimal LpLp discrepancy are known for 1<p<∞1<p<∞, much less is known about the L∞L∞ discrepancy. In this talk we describe recent advances in the study of the bounded mean oscillation and exponential Orlicz norm of the discrepancy function.