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Transactions of the American Mathematical Society

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Systems of diagonal Diophantine inequalities

Author: Eric Freeman
Journal: Trans. Amer. Math. Soc. 355 (2003), 2675-2713
MSC (2000): Primary 11D75; Secondary 11D41, 11D72, 11P55
Published electronically: March 17, 2003
MathSciNet review: 1975395
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Abstract: We treat systems of real diagonal forms $F_1({\mathbf x}), F_2({\mathbf x}), \ldots, F_R({\mathbf x})$ of degree $k$, in $s$ variables. We give a lower bound $s_0(R,k)$, which depends only on $R$ and $k$, such that if $s \geq s_0(R,k)$ holds, then, under certain conditions on the forms, and for any positive real number $\epsilon$, there is a nonzero integral simultaneous solution $\displaystyle{{\mathbf x}\in {\mathbb Z}^s}$ of the system of Diophantine inequalities $\vert F_i({\mathbf x})\vert < \epsilon$ for $1 \leq i \leq R$. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.

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  • 1. M. Aigner, Combinatorial theory, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York/Heidelberg/Berlin, 1979. MR 80h:05002
  • 2. R. C. Baker, Diophantine inequalities, London Mathematical Society Monographs, New Series, 1, The Clarendon Press, Oxford University Press, New York, 1986. MR 88f:11021
  • 3. V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms, Ann. of Math. (2) 150 (1999), no. 3, 977-1027. MR 2001b:11087
  • 4. B. J. Birch and H. Davenport, Indefinite quadratic forms in many variables, Mathematika, 5 (1958), 8-12. MR 20:3104
  • 5. J. Brüdern and R. J. Cook, On simultaneous diagonal equations and inequalities, Acta Arith. 62 (1992), 125-149. MR 93h:11036
  • 6. H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, Ann Arbor Publishers, Ann Arbor, MI, 1963. MR 28:3002
  • 7. H. Davenport and D. J. Lewis, Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557-595. MR 39:6848
  • 8. J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bureau Standards 69B (1965), 67-72. MR 32:7441
  • 9. D. E. Freeman, A note on one cubic Diophantine inequality, J. London Math. Soc. (2), 61 (2000), no.1, 25-35. MR 2001c:11043
  • 10. -, Quadratic Diophantine inequalities, J. Number Theory, 89 (2001), no.2, 268-307.
  • 11. -, Asymptotic lower bounds for Diophantine inequalities, to appear in Mathematika.
  • 12. -, Asymptotic lower bounds and formulas for Diophantine inequalities, to appear in the Proceedings of the Millennial Conference in Number Theory.
  • 13. L. Low, J. Pitman and A. Wolff, Simultaneous diagonal congruences, J. Number Theory 29 (1988), 31-59. MR 89g:11030
  • 14. I. D. Meir, Simultaneous diagonal $p$-adic equations, Mathematika 45 (1998), 337-349. MR 2000k:11052
  • 15. T. Nadesalingam and J. Pitman, Bounds for solutions of simultaneous diagonal equations of odd degree, Théorie des nombres (Québec, PQ, 1987), 703-734, de Gruyter, Berlin, 1989. MR 91f:11021
  • 16. -, Simultaneous diagonal inequalities of odd degree, J. Reine Angew. Math., 394 (1989), 118-158. MR 91c:11019
  • 17. W. M. Schmidt, Diophantine inequalities for forms of odd degree, Adv. Math. 38 (1980), 128-151. MR 82h:10033
  • 18. -, Simultaneous rational zeros of quadratic forms, Seminar Delange-Pisot-Poitou 1981. Progress in Math., Vol. 22, Birkhäuser, Boston, MA, 1982, 281-307. MR 84g:10041
  • 19. R. C. Vaughan, On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122-170. MR 87j:11103
  • 20. -, On Waring's problem for smaller exponents. II, Mathematika 33 (1986), 6-22. MR 87j:11104
  • 21. -, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, Cambridge, U.K., 1997. MR 98a:11133
  • 22. Y. Wang, Diophantine equations and inequalities in algebraic number fields, Springer-Verlag, Berlin/Heidelberg/New York, 1991. MR 92a:11036
  • 23. T. D. Wooley, New estimates for smooth Weyl sums, J. London Math. Soc. (2) 51 (1995), 1-13. MR 96e:11109

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Additional Information

Eric Freeman
Affiliation: Department of Mathematics, University of Colorado, 395 UCB, Boulder, Colorado 80309
Address at time of publication: School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540

Keywords: Combined systems of Diophantine equations and inequalities, forms in many variables, applications of the Hardy-Littlewood method.
Received by editor(s): October 15, 2001
Published electronically: March 17, 2003
Additional Notes: The author was supported by an NSF Postdoctoral Fellowship.
Article copyright: © Copyright 2003 American Mathematical Society

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