Vol. 29, No. 1, 1969

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Irregularities of distribution. III

Wolfgang M. Schmidt

Vol. 29 (1969), No. 1, 225–234

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 (A), where μ denotes the measure which is normalized so that μ(S) = 1. Hence define the discrepancy Δ(A) by

Δ (A) = |ν(A)− N μ(A)|.

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.

Mathematical Subject Classification
Primary: 10.33
Received: 5 December 1967
Published: 1 April 1969
Wolfgang M. Schmidt