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Effective estimates on indefinite ternary forms

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Abstract

We give an effective proof of a theorem of Dani and Margulis regarding values of indefinite ternary quadratic forms at primitive integer vectors. The proof uses an effective density-type result for orbits of the groups SO(2, 1) on SL(3, ℝ)/SL(3, ℤ).

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References

  1. V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms, Annals of Mathematics 150(1999), 977–1027.

    Article  MATH  MathSciNet  Google Scholar 

  2. E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, Vol. 4, Cambridge University Press, Cambridge, 2006.

    MATH  Google Scholar 

  3. J. W. S. Cassels, Rational Quadratic Forms, London Mathematical Society Monographs, Vol. 13, Academic Press, London, 1978.

    MATH  Google Scholar 

  4. J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physiscal and Engineering Sciences 248 (1955), 73–96.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Inventiones Mathematica 47 (1978), 101–138.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. G. Dani, On uniformly distributed orbits of certain horocycle flows, Ergodic Theory and Dynamical Systems 2 (1982), 139–158.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. G. Dani, On orbits of unipotent flows on homogeneous spaces. II, Ergodic Theory and Dynamical Systems 6 (1986), 167–182.

    MATH  MathSciNet  Google Scholar 

  8. S. G. Dani, A proof of Margulis’ theorem on values of quadratic forms, independent of the axiom of choic, L’Enseignement Mathématique. Revue Internationale 40 (1994), 49–58.

    MATH  MathSciNet  Google Scholar 

  9. H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, Journal of the London Mathematical Society 21 (1946), 185–193.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. G. Dani and G.A. Margulis, Values of quadratic forms at primitive integral points, Inventiones Mathematicae 98 (1989), 405–424.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of SL(3,R), Mathematische Annalen 286 (1990), 101–128.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. G. Dani and G. A. Margulis, Values of quadratic forms at integral points: an elementary approach, L’Enseignement Mathématique. Revue Internationale 36 (1990), 143–174.

    MATH  MathSciNet  Google Scholar 

  13. S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel’fand Seminar, Advances in Soviet Mathematics, Vol. 16, American Mathematical Society, Providence, RI, 1993, pp. 91–137.

    Google Scholar 

  14. M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh, The distribution of periodic torus orbits on homogeneous spaces, Duke Mathematical Journal 148 (2009), 119–174.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Inventiones Mathematicae 177 (2009), 137–212.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Götze and G. A. Margulis, Distribution of Values of Quadratic Forms at Integral Points, ArXiv e-prints, April 2010.

  17. D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Annals of Mathematics 148 (1998), 339–360.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. A. Margulis, The action of unipotent groups in a lattice space, Matematicheskiĭ Sbornik 86 no. 128 (1971), 552–556.

    MathSciNet  Google Scholar 

  19. G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homog`enes, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 304 (1987), 249–253.

    MATH  MathSciNet  Google Scholar 

  20. G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 377–398.

    Chapter  Google Scholar 

  21. G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Mathematical Society of Japan, Tokyo, 1991, pp. 193–215.

    Google Scholar 

  22. A. Mohammadi, A special case of effective equidistribution with explicit constants, Ergodic Theory and Dynamical Systems 32 (2012), 237–247.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Inventiones Mathematicae 116 (1994), 347–392.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. A. Margulis and G. M. Tomanov, Measure rigidity for almost linear groups and its applications, Journal d’Analyse Mathématique 69 (1996), 25–54.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Ratner, Horocycle flows, joinings and rigidity of products, Annals of Mathematics 118 (1983), 277–313.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Mathematica 165 (1990), 229–309.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Inventiones Mathematicae 101 (1990), 449–482.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Ratner, On Raghunathan’s measure conjecture, Annals of Mathematics 134 (1991), 545–607.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Mathematical Journal 63 (1991), 235–280.

    Article  MATH  MathSciNet  Google Scholar 

  30. N. A. Shah, Unipotent flows on products of SL(2,K)/Γ’s, in Dynamical Systems and Diophantine Approximation, Séminaires et Congrès, Vol. 19, Société Math ematique de France, Paris, 2009, pp. 69–104

    Google Scholar 

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Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Elon Lindenstrauss

  2. Yale University Mathematics Department, PO Box 208283, New Haven, CT, 06520, USA

    Gregory Margulis

Authors
  1. Elon Lindenstrauss
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  2. Gregory Margulis
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Corresponding author

Correspondence to Elon Lindenstrauss.

Additional information

Dedicated to the memory of Joram Lindenstrauss

E. L. was supported in part by the ERC (AdG Grant 267259), the ISF (983/09) and the NSF (DMS-0800345).

G. M. was supported in part by the NSF (DMS-0801195 and DMS-1265695).

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Lindenstrauss, E., Margulis, G. Effective estimates on indefinite ternary forms. Isr. J. Math. 203, 445–499 (2014). https://doi.org/10.1007/s11856-014-1110-3

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  • DOI: https://doi.org/10.1007/s11856-014-1110-3

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