Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T05:12:10.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Note on irregularities of distribution

Published online by Cambridge University Press:  26 February 2010

H. Davenport
H. Davenport
Affiliation:
University College, London.
Get access

Extract

Let (x1, y1), …, (xN, yN) be N points in the square 0 ≤ x < 1, 0 ≤ y < 1. For any point (ξ, η) in this square, let S(ξ, η) denote the number of points of the set satisfying

Type
Research Article
Information
Mathematika , Volume 3 , Issue 2 , December 1956 , pp. 131 - 135
Copyright
Copyright © University College London 1956

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

page 131 note * Mathematika, 1 (1954), 73–79.CrossRefGoogle Scholar

page 131 note † It was proved by Lerch in 1904 that, with this construction,

see Koksma, Diophantische Approximationen, Kap IX, (17). It remains an unsolved problem whether or not there exists a set of points for which a better estimate holds.

page 132 note * For an account of this problem, which was proposed some years ago by Prof. J. E. Littlewood, see Cassels, J. W. S. and Swinnerton-Dyer, H. P. F., Phil. Trans. Royal Soc. A, 248 (1955), 73–96.CrossRefGoogle Scholar

page 133 note * We use Vinogradov's symbolism F ≪ G as an equivalent for F = O(G).

page 134 note * For a proof, see Cassels and Swinnerton-Dyer, loc. cit., Lemma 5.