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Simultaneous approximation of two additive forms

Published online by Cambridge University Press:  24 October 2008

Ming-Chit Liu
Ming-Chit Liu
Affiliation:
Department of Mathematics, University of Hong Kong, Hong Kong, B.C.C.
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Extract

1. Introduction. In 1948, H. Heibronn (6) improved Vinogradov's result (11) and obtained that for any ε > 0 there exists some positive constant C(ε) which depends on ε only such that, for any real number θ and any integer N ≥ 1, there is an integer x satisfying

Here ∥t∥ means the distance from t to the nearest integer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 77 - 82
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Cook, R. J.The fractional parts of an additive form. Proc. Cambridge Philos. Soc. 72 (1972), 209–212.CrossRefGoogle Scholar
(2)Danicic, I.An extension of a theorem of Heilbronn. Mathematika 5 (1958), 30–37.CrossRefGoogle Scholar
(3)Danicic, I.The distribution (mod 1) of pairs of quadratic forms with integer variables. J. London Math. Soc. 42 (1967), 618–623.CrossRefGoogle Scholar
(4)Davenport, H.On a theorem of Heilbronn. Quart. J. Math. Oxford, Ser. 2, 18 (1967), 339–344.CrossRefGoogle Scholar
(5)Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers (Oxford, 4th ed., 1960).Google Scholar
(6)Heilbronn, H.On the distribution of the sequence θn 2 (mod 1). Quart. J. Math. Oxford 19 (1948), 249–256.CrossRefGoogle Scholar
(7)Hua, L. K.Additive Theory of Prime Numbers. Translations of Mathematical Monographs, vol. 13 (Amer. Math. Soc. Providence R.I., 1965).Google Scholar
(8)Landau, E.Vorlesungen über Zahlentheorie, Bd. I (Leipzig, Hirzel, 1927).Google Scholar
(9)Liu, M. C.On the fractional parts of θnk and ϕnk. Quart. J. Math. Oxford, Ser. 2, 21 (1970), 481–486.CrossRefGoogle Scholar
(10)Liu, M. C.On a theorem of Heilbronn concerning the fractional part of θn 2. Canad. J. Math. 22 (1970), 784–788.CrossRefGoogle Scholar
(11)Vinogradov, I. M.Analytischer Beweis des Satzes über die Verteilung der Bruchteile eines ganzen Polynoms. Bull. Acad. Sci. USSR (6) 21 (1927), 567–578.Google Scholar
(12)Vinogradov, I. M.The Method of Trigonometric Sums in the Theory of Numbers (New York, Interscience, 1954).Google Scholar