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Cubic Diophantine inequalities

  • Jörg Brüdern (a1)

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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 λij, 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.

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References

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1.Baker, R. C.. Cubic diophantine inequalities. Mathematika, 29 (1982), 8392.
2.Baker, R. C. and Harman, G.. Diophantine inequalities with mixed powers. J. Number Theory, 18 (1984), 6985.
3.Brüdern, J.. Addtive diophantine inequalities with mixed powers I. Mathematika, 34 (1987), 124130.
4.Brüdern, J.. Addtive diophantine inequalities with mixed powers II. Mathematika, 34 (1987), 131142.
5.Davenport, H.. On Waring's problem for cubes. Ada Math., 71 (1939), 123143.
6.Davenport, H.. On indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81101.
7.Davenport, H. and Roth, K. F.. The solubility of certain diophantine inequalities. Mathematika, 2 (1955), 8196.
8.Liu, M. C., Ng, S.-M. and Tsang, K. M.. An improved estimate for certain diophantine inequalities. Proc. Amer. Math. Soc., 78 (1980), 457463.
9.Vaughan, R. C.. The Hardy-Littlewood method (University Press, Cambridge, 1981).
10.Vaughan, R. C.. Sums of three cubes. Bull. London Math. Soc., 17 (1985), 1720.
11.Vaughan, R. C.. On Waring's problem for cubes. J. reine angew. Math., 365 (1986), 122170.
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Cubic Diophantine inequalities

  • Jörg Brüdern (a1)

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