Quasirandom points and global function fields

Abstract

ABSTRACT. We present a survey of recent work of the authors in which sequences of quasirandom points are constructed by new methods based on global function fields. These methods yield significant improvements on all earlier constructions. The most powerful of these methods employ global function fields with many rational places, or equivalently algebraic curves over finite fields with many rational points. With the help of class field theory for global function fields, it can be shown that our constructions are best possible in the sense of the order of magnitude of quality parameters. The paper contains also a new construction of sequences of quasirandom points and new facts about the earlier constructions designed by the authors. Introduction The motivation for the work that we want to present here stems from the theory of uniform distribution of sequences in number theory and from quasi-Monte Carlo methods in numerical analysis. A key problem in these areas is how to distribute points as uniformly as possible over an s -dimensional unit cube I s = [0, 1] s , s ≥ 1. A precise formulation of this problem will be given below. The essence of our work is that methods based on global function fields (or, equivalently, on algebraic curves over finite fields) yield excellent constructions of (finite) point sets and (infinite) sequences with strong uniformity properties. In fact, these methods are so powerful that they lead to constructions which are, in a sense to be explained later, best possible.