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Note on irregularities of distribution

Published online by Cambridge University Press:  26 February 2010

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

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

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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.