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Cubic Diophantine inequalities

Published online by Cambridge University Press:  26 February 2010

Jörg Brüdern
Jörg Brüdern
Affiliation:
Geismar Landstrasse 97, 3400 Göttingen, Federal Republic of Germany.
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Extract

It was shown by Davenport and Roth [7] that the values taken by

at integer points ( x1, …, x8) ∈ ℤ8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang [8] showed that for all the inequality

has infinitely many solutions in integers. Later Baker [1] obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable.

Type
Research Article
Information
Mathematika , Volume 35 , Issue 1 , June 1988 , pp. 51 - 58
Copyright
Copyright © University College London 1988

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References

1.Baker, R. C.. Cubic diophantine inequalities. Mathematika, 29 (1982), 83–92.CrossRefGoogle Scholar
2.Baker, R. C. and Harman, G.. Diophantine inequalities with mixed powers. J. Number Theory, 18 (1984), 69–85.CrossRefGoogle Scholar
3.Brüdern, J.. Addtive diophantine inequalities with mixed powers I. Mathematika, 34 (1987), 124–130.CrossRefGoogle Scholar
4.Brüdern, J.. Addtive diophantine inequalities with mixed powers II. Mathematika, 34 (1987), 131–142.CrossRefGoogle Scholar
5.Davenport, H.. On Waring's problem for cubes. Ada Math., 71 (1939), 123–143.Google Scholar
6.Davenport, H.. On indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
7.Davenport, H. and Roth, K. F.. The solubility of certain diophantine inequalities. Mathematika, 2 (1955), 81–96.CrossRefGoogle Scholar
8.Liu, M. C., Ng, S.-M. and Tsang, K. M.. An improved estimate for certain diophantine inequalities. Proc. Amer. Math. Soc., 78 (1980), 457–463.CrossRefGoogle Scholar
9.Vaughan, R. C.. The Hardy-Littlewood method (University Press, Cambridge, 1981).Google Scholar
10.Vaughan, R. C.. Sums of three cubes. Bull. London Math. Soc., 17 (1985), 17–20.CrossRefGoogle Scholar
11.Vaughan, R. C.. On Waring's problem for cubes. J. reine angew. Math., 365 (1986), 122–170.Google Scholar