This paper deals with
irregularities of distribution on spheres. Suppose there are N points on the unit
sphere S = Sn of Euclidean En+1. If these points are reasonably well distributed one
would expect that for every simple measurable subset A of the sphere the number
ν(A) of these points in the subset is fairly close to Nμ(A), where μ denotes the
measure which is normalized so that μ(S) = 1. Hence define the discrepancy Δ(A)
by
(1)
It is shown in the present paper that there are very simple sets A, namely
intersections of two half spheres, for which Δ(A) is large. This result is analogous to
a theorem of K. F. Roth concerning irregularities of distribution in an n-dimensional
cube.
