Introduction

Abstract

In this chapter, we introduce the concept of discrepancy. We formulate a basic problem concerning discrepancy for rectangles, we show its connections to the discrepancy of infinite sequences in the unit interval, and we briefly comment on the historical roots of discrepancy in the theory of uniform distribution (Section 1.1). In Section 1.2, we introduce discrepancy in a general geometric setting, as well as some variations of the basic definition. Section 1.3 defines discrepancy in a seemingly different situation, namely for set systems on finite sets, and shows a close relationship to the previously discussed “Lebesguemeasure” discrepancy. Finally, Section 1.4 is a mosaic of notions, results, and comments illustrating the numerous and diverse connections and applications of discrepancy theory. Most of the space in that section is devoted to applications in numerical integration and similar problems, which by now constitute an extensive branch of applied mathematics, with conventions and methods quite different from “pure” discrepancy theory.