Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T01:01:54.875Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

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