IRREGULARITIES OF DISTRIBUTION .3.

This paper deals with irregularities of distribution on spheres. Suppose there are N points on the unit sphere S = Sn of Euclidean. If these points are reasonably well distributed one would expect that for every simple measurable subset A of the sphere the number v(A) of these points in the subset is fairly close to Nμ(A), where π denotes the measure which is normalized so that μ(S) = 1. Hence define the discrepancy △(A) by It is shown in the present paper that there are very simple sets A, namely intersections of two half spheres, for which △(A) is large. This result is analogous to a theorem of K. F. Roth concerning irregularities of distribution in an n-dimensional cube. © 1969 by Pacific Journal of Mathematics.