Vol. 29, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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