Principles of a new method in the study of irregularities of distribution

Abstract

The concept of irregularity o f distribution (or discrepancy) is difficult to define precisely, but its spirit is much easier to describe because it springs directly from Hermann Weyl's influential paper [W]. Here Weyl discusses uniform distribution of sequences in the unit interval. Among other things, he develops several of the standard criteria for what is now known as the weak convergence of measures, including the test that now defines this important concept. Weyl was concerned with the weak convergence of sequences of atomic measures to Lebesgue measure. The study of discrepancy revolves about the problem of estimating how rapidly a sequence of atomic measures can converge to a given continuous measure. Included is the closely related second problem of determining how well an atomic measure with n freely placed atoms can approximate a given continuous measure. The classical problem of discrepancy as posed by Van der Corput concerns the unit interval. If r i is a fixed sequence in [0, 1 ], define the signed measure v, = v + v~by letting v + be Lebesgue measure and v~ be the atomic measure that assigns weight l/n to r i for i<n. The problem is to estimate how small nD(n) can remain ifD(n) is defined by