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The fractional parts of an additive form

Published online by Cambridge University Press:  24 October 2008

R. J. Cook
R. J. Cook
Affiliation:
University College, Cardiff
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Extract

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that

,

where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 2 , September 1972 , pp. 209 - 212
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Danicic, I. Ph.D. Thesis (London, 1957).Google Scholar
(2)Danicic, I.An extension of a theorem of Heilbronn, Mathematika. 5 (1958), 30–37.CrossRefGoogle Scholar
(3)Davenport, H.Analytic methode for Diophantine equatons and Diophantine inequalities. (Ann Arbor, Michigan, 1962).Google Scholar
(4)Davenport, H.On a theorem of Heilbronn. Quart. J. Math. Oxford (2). 18 (1967), 339–344.CrossRefGoogle Scholar
(5)Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers, 4th ed. (Oxford, 1965).Google Scholar
(6)Heilbronn, H.On the distribution of the sequence n 2 θ (mod 1). Quart. J. Math. Oxford. 19 (1948), 249–256.CrossRefGoogle Scholar
(7)Vinogradov, I. M.The method of trigonometrical ume in the theory of numbers (translated by Roth, K. F. and Davenport, Anne) (Interscience Publishers, 1954).Google Scholar