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Cubic Diophantine inequalities

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

R. C. Baker ,
J. Brüdern  and
T. D. Wooley
R. C. Baker
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A.
J. Brüdern
Affiliation:
Mathematisches Institut A, Postfach 80-11-40, Universitàt Stuttgart, D-7051, Stuttgart, Germany.
T. D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003, U.S.A.
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Extract

Let λ1, …, λs be nonzero real numbers and suppose that λ1/λs is irrational. In 1955, Davenport and Roth showed [6] that the values taken by

at integer points (x1, …, xs) are dense on the real line, provided that s≥8. In the present paper we obtain the same result with seven variables.

MSC classification

Secondary: 11D75: Diophantine inequalities
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 264 - 277
Copyright
Copyright © University College London 1995

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References

1.Baker, R. C.. Cubic Diophantine inequalities. Mathematika, 29 (1982), 83–92.Google Scholar
2.Brüdern, J.. Additive Diophantine inequalities with mixed powers I. Mathematika, 34 (1987), 124–130.Google Scholar
3.Brüdern, J.. Cubic Diophantine inequalities. Mathematika, 35 (1988), 51–58.Google Scholar
4.Brüdern, J.. A note on cubic exponential sums. Sém. Théorie des Nombres, Paris, 1990–1991, 23–24, (S. David, ed.), Progr. Math.Google Scholar
5.Davenport, H.. On indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
6.Davenport, H. and Roth, K. F.. The solubility of certain Diophantine inequalities. Mathematika, 2 (1955), 81–96.Google Scholar
7.Linnik, Ju. V.. On the representation of large numbers as sums of seven cubes. Mat. Sbornik, 12 (1943), 218–224.Google Scholar
8.Vaughan, R. C.. The Hardy-Littlewood method (Cambridge University Press, 1981).Google Scholar
9.Vaughan, R. C.. Some remarks on Weyl sums. Colloq. Math. Soc. Janos Bolyai, 34 (Elsevier, North-Holland, Amsterdam 1984), 1585–1602.Google Scholar
10.Vaughan, R. C.. On Waring's problem for cubes. J. Reine Angew. Math., 365 (1986), 122–170.Google Scholar
11.Vaughan, R. C.. A new iterative method in Waring's problem, Ada Math., 162 (1989), 1–71.Google Scholar
12.Vaughan, R. C.. On Waring's problem for cubes II. J. London Math. Soc. (2), 39 (1989), 205–218.CrossRefGoogle Scholar
13.Wooley, T. D.. Breaking classical convexity in Waring's problem: sums of cubes and quasidiagonal behaviour. Inventiones Math., to appear.Google Scholar