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Asymptotic lower bounds for Diophantine inequalities

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

D. Eric Freeman
D. Eric Freeman
Affiliation:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309-0395, U.S.A.
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Extract

§1. Introduction. In 1946, Davenport and Heilbronn [9] proved a result which opened up the study of Diophantine inequalities. Suppose that Q(x) is a diagonal quadratic form with non-zero real coefficients in s variables. We write

MSC classification

Secondary: 11D75: Diophantine inequalities
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 127 - 159
Copyright
Copyright © University College London 2000

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References

1.Baker, R. C.. Diophantine Inequalities. London Mathematical Society Monographs, New Series, 1, (Oxford University Press, New York, 1986).Google Scholar
2.Baker, R. C., Brüdern, J. and Wooley, T. D.. Cubic Diophantine inequalities. Mathematika, 42 (1995), 264–277.Google Scholar
3.Bentkus, V. and Götze, F.. Lattice point problems and distribution of values of quadratic forms. Ann. of Math. 150 (1999), 977–1027.Google Scholar
4.Birch, B. J.. Homogeneous forms of odd degree in a large number variables. Mathematika, 4 (1957), 102–105.Google Scholar
5.Brüdern, J. and Cook, R. J.. On simultaneous diagonal equations and inequalities. Acta Arith., 62 (1992), 125–149.Google Scholar
6.Davenport, H.. Indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
7.Davenport, H.. Analytic Methods for Diophantine Equations and Diophantine Inequalities (Ann Arbor Publishers, Ann Arbor, 1962).Google Scholar
8.Davenport, H.. Cubic forms in sixteen variables. Proc. Roy. Soc. Ser. A, 272 (1963), 285–303.Google Scholar
9.Davenport, H. and Heilbronn, H.. On indefinite quadratic forms in five variables. J. London Math. Soc., 21 (1946), 185–193.Google Scholar
10.Davenport, H. and Roth, K. F.. The solubility of certain Diophantine inequalities. Mathematika, 2 (1955), 81–96.CrossRefGoogle Scholar
11.Eskin, A., Margulis, G. A. and Mozes, S.. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. of Math. (2), 147 (1998), 93–141.Google Scholar
12.Freeman, D. E.. Quadratic Diophantine Inequalities. J. Number Theory. 89 (2001), 268–307.Google Scholar
13.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”, II: Proof that every large number is the sum of at most 21 biquadrates. Math. Z., 9 (1921), 14–27.Google Scholar
14.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”, IV: The singular series in Waring's problem. Math. Z., 12 (1922), 161–188.Google Scholar
15.Margulis, G. A.. Discrete subgroups and ergodic theory. Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987). Academic Press (Boston, 1989), pp. 377–398.Google Scholar
16.Schmidt, W. M.. Simultaneous rational zeros of quadratic forms. Seminar Delange-Pisot-Poitou 1981. Prog, in Math., 22 (1982), 281–307.Google Scholar
17.Schmidt, W. M.. On cubic polynomials IV: Systems of rational equations. Monatsh. Math., 93 (1982), 329–348.Google Scholar
18.Schmidt, W. M.. The density of integer points on homogeneous varieties. Acta Math., 154 (1985), no. 3–4, 243–296.Google Scholar
19.Vaughan, R. C.. On Waring's problem for cubes. J. Reine Angew. Math., 365 (1986), 122–170.Google Scholar
20.Vaughan, R. C.. On Waring's problem for smaller exponents, II. Mathematika, 33 (1986), 6–22.Google Scholar
21.Vaughan, R. C.. A new iterative method in Waring's problem. Acta Math., 162 (1989), 1–71.Google Scholar
22.Vaughan, R. C.. The Hardy-Liltlewood Method (2nd edn.). (Cambridge University Press, Cambridge, 1997).Google Scholar
23.Walfisz, A.. Über Gitterpunkte in mehrdimensionalen Ellipsoiden (Dritte Abhandlung). Math. Z., 27 (1927), 245–268.CrossRefGoogle Scholar
24.Wooley, T. D.. New estimates for smooth Weyl sums. J. London Math. Soc. (2), 51 (1995), 1–13.Google Scholar