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Small fractional parts of quadratic forms in many variables

Published online by Cambridge University Press:  26 February 2010

R. J. Cook
R. J. Cook
Affiliation:
University of Sheffield
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Extract

Let

be a real quadratic form in n variables x1,…, xn and let ‖θ‖ denote the distance from θ to the nearest integer. Danicic [3] proved that if N > 1 and ξ > 0 then there exist integers x1,…, xn, not all zero, satisfying

where C depends on n and ξ only.

Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 25 - 29
Copyright
Copyright © University College London 1980

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References

1.Birch, B. J. and Davenport, H.. “Indefinite quadratic forms in many variables”, Mathematikca, 5 (1958), 8–12.CrossRefGoogle Scholar
2.Cassels, J. W. S.. An introduction to Diophantine approximation (Cambridge University Press, 1965).Google Scholar
3.Danicic, I.. “An extension of a theorem of Heilbronn”, Mathematika, 5 (1958), 30–37.CrossRefGoogle Scholar
4.Davenport, H.. “Indefinite quadratic forms in many variables”, Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
5.Schlickewei, H. P.. “On indefinite diagonal forms in many variables”, J. Reine Angew. Math., 307/8 (1979), 279–294.Google Scholar