Upper bounds in irregularities of distribution

The subject of irregularities of distribution arises from uniform distribution, but is of independent interest, and owes its current prominance to the fundamental contribution of K.F. Roth [23–27] and W.M. Schmidt [28–38]. While the theory of uniform distribution may be described as qualitative, the theory of irregularities of distribution is definitely quantitative in nature, as one seeks to measure (with great precision in many instances) the actual discrepancy (in a certain sense) incurred by a finite set of points distributed within a finite region. There are lower bound results which say that the discrepancy of a set of points cannot be less than a certain minimum value which only depends on the number of points in question, and not where they are placed within the finite region. On the other hand, there are upper bound results which say that if the points are placed carefully, then the discrepancy cannot exceed a certain maximum value which again only depends on the number of points in question. In many instances, it has been shown that this upper bound is a constant multiple of the lower bound. The tools in this subject are diverse, and involve ideas in harmonic analysis, number theory, geometry, combinatorics and probability theory. The purpose of this paper is to discuss some of the central ideas in the study of upper bounds in the theory of irregularities of distribution. This paper is not intended as a survey, and many results have been omitted. Also, only a few proofs are given in detail; in many other instances, we shall discuss briefly the main ideas and omit the (often very complicated) details. In §1, we shall give an overview of the subject as a whole, and illustrate its development from its infancy to the present day by mentioning some of the key results. We also mention many extremely difficult problems which remain unsolved. In §§2–6, we shall discuss the main ideas in the study of upper bound questions. We conclude this paper by proving in the appendix the famous lower bound result of Roth [23] which laid the foundations of the subject. The material in this paper is the subject of a series of lectures given at Macquarie University in the first half of 1992. I would like to express my thanks to Grigori