Uniformity and Irregularity

Introduction. The purpose of this report is to draw attention to some nontrivial connections between the ("continuous") theory of irregularities of distribution and discrete mathematics. The object of the theory of irregularities of distribution is to measure the uniformity (or nonuniformity) of sequences and point sets. For instance: how uniformly can an arbitrary set of N points in the unit cube be distributed relative to a given family of "nice" sets (e.g., boxes with sides parallel to the coordinate axes, rotated boxes, balls, all convex sets). This theory lies on the border of many branches of mathematics (number theory, discrete geometry, combinatorics, etc.) and has very important applications, e.g., in numerical integration. Here we focus, of course, on the combinatorial aspects of the theory. As an illustration, we shall discuss first two problems of a discrete nature which have fascinating connections with this "continuous" theory. The first question is concerned with balanced two-colorings of finite sets in a square (the problem is due to G. Tusnâdy). Let P = { p i , . . . ,PJV} be a distribution of N points in the unit square [0, l ) 2 . Let / : P —• {—1, +1} be a "two-coloring" of P. Let B denote any rectangle in [0, l ) 2 with sides parallel to the coordinate axes (an aligned rectangle, in short). Consider the function