The Great Open Problem in Discrepancy Theory

Abstract: 

Consider the supremum norm of the discrepancy function with respect to aligned rectangular boxes. In the 22-dimensional case, the precise order of magnitude was determined over forty years ago. In higher dimensions, there remains a significant gap between the best known lower and upper bounds. Since the 1980s, this has been known as the Great Open Problem, and is arguably the most famous unsolved problem in geometric discrepancy theory.

Consider the corresponding L1L1-norm. In the 22-dimensional case, the precise order of magnitude was determined over thirty years ago. In higher dimensions, there again remains a significant gap between the best known lower and upper bounds. This problem is considered to be equally hard as the Great Open Problem.

We now turn to the corresponding LqLq-averages. For every q>1q>1, the precise order of magnitude was determined over thirty years ago for every dimension. However, absolutely nothing non-trivial is known for any qq satisfying 0<q<10<q<1.

Indeed, experts in the subject are somewhat divided on the likely outcome of the Great Open Problem.

In this talk, we shall describe some very recent results. On the one hand, we show that one of the likely outcomes of the Great Open Problem is true on average. On the other hand, we describe another result that supports the other likely outcome of the Great Open Problem.