Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T05:26:46.762Z Has data issue: false hasContentIssue false
  • English
  • Français

Cubic Diophantine inequalities

Published online by Cambridge University Press:  26 February 2010

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey.
Get access

Extract

Let λ1, …, λ8 be any non-zero real numbers not all of the same sign and not all in rational ratios. According to a theorem of Davenport and Roth [3], given a real number κ, the inequality

has infinitely many solutions in positive integers for any ε > 0. Recently Liu, Ng and Tsang [5] gave a refinement of this result: for any δ > 0, the inequality

has infinitely many solutions in positive integers. In the present note we obtain a better exponent.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 83 - 92
Copyright
Copyright © University College London 1982

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

1.Davenport, H.. On Waring's problem for cubes. Acta Mathematica, 71 (1939), 123–143.CrossRefGoogle Scholar
2.Davenport, H.. Indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
3.Davenport, H. and Roth, K. F.. The solubility of certain Diophantine inequalities. Mathematika, 2 (1955), 81–96.CrossRefGoogle Scholar
4.Hua, L.-K.. On Waring's problem. Quart. J. Math. (Oxford), 9 (1938), 199–202.CrossRefGoogle Scholar
5.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
6.Pitman, J. and Ridout, D.. Diagonal cubic equations and inequalities. Proc. Roy. Soc. A, 297 (1967), 476–502.Google Scholar
7.Watson, G. L.. On indefinite quadratic forms in five variables. Proc. London Math. Soc. (3), 3 (1953), 170–181.CrossRefGoogle Scholar