My main research interest is in the theory of irregularities of point distribution. The subject originates from the theory of uniform distribution, but is of independent interest, and owes its current prominance to the fundamental contribution of Klaus Roth and Wolfgang Schmidt, two of the greatest mathematicians of the twentieth century. It has flourished over the past thirty years due to the work of a small group of about ten to twenty active researchers. Having been introduced to this subject by Roth, I am very fortunate to be in this happy group, the other members of which are all overseas. I am particularly lucky to be able to enjoy longstanding collaborations with my great friends József Beck, Maxim Skriganov and Giancarlo Travaglini.
One of the major attractions of this subject is the nature of the fundamental questions involved. We want to study this subject because it is very interesting, very difficult and not many people work on it.
While the theory of uniform distribution may be described as qualitative, the theory of irregularities of point 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 currently involve ideas in harmonic analysis, algebra, number theory, geometry, combinatorics, probability theory, graph theory and coding theory.
In the case of lower bounds, the general idea is very simple. However, the implementation is hard. To understand this, note that the expectation is a real number, not necessarily an integer, while the actual number of points that fall into any sub-region of the finite region is clearly an integer. It follows that many sub-regions have non-zero discrepancies (by discrepancy, we mean the difference between the actual number of points in the sub-region and the expectation for that same sub-region). It follows that one should find ways of combining such non-zero discrepancies to obtain a result that there must be some sub-region with large discrepancy. The difficulty is, of course, that these non-zero discrepancies may be positive or negative, so that there may be a lot of cancellation if we attempt to add them together. The study of lower bound questions is therefore a study of ways of getting round this serious handicap. Such ways involve ideas in harmonic analysis, elementary geometry, integral geometry, probability theory and graph theory.
For upper bounds, there is no general simple idea. Given any situation, the aim here is to find an essentially best possible distribution. There are arguments which involve very complicated and delicate combinatorial constructions, followed by powerful probabilistic techniques. There are arguments which involve a great deal of graph theory. There are also arguments which involve very simple constructions followed by a careful analysis of the Fourier series of the discrepany functions that arise. More recently, ideas from coding theory have been used in an effective way to study some of these problems too.
For a discussion of the development of this subject up to the mid-1980's, please see the monograph by J. Beck and W.W.L. Chen, Irregularities of Distribution (Cambridge Tracts in Mathematics 89, Cambridge University Press, 1987). For a discussion of more recent developments and to get a wider perspective, please see the beautifully written monograph by J. Matousek, Geometric Discrepancy (Algorithms and Combinatorics 18, Springer Verlag, 1999).