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A quantitative Oppenheim theorem for generic ternary quadratic forms
Values of random polynomials at integer points
1. | Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA |
2. | Department of Mathematics, Yale University, Box 208283, New Haven, CT 06520, USA |
Using classical results of Rogers [
References:
[1] |
J. S. Athreya,
Random affine lattices, Contemp. Math., 639 (2015), 160-174.
|
[2] |
J. S. Athreya and G. A. Margulis,
Logarithm laws for unipotent flows. I, J. Mod. Dyn., 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
[3] |
J. N. Bernstein,
Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen., 6 (1972), 26-40.
|
[4] |
J. Bourgain,
A quantitative Oppenheim theorem for generic diagonal quadratic forms, Israel J. Math., 215 (2016), 503-512.
doi: 10.1007/s11856-016-1385-7. |
[5] |
A. Chambert-Loir and Y. Tschinkel,
Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math., 2 (2010), 351-429.
doi: 10.1142/S1793744210000223. |
[6] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.
doi: 10.2307/120984. |
[7] |
A. Eskin, G. Margulis and S. Mozes,
Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2-tori, Ann. of Math., 161 (2005), 679-725.
doi: 10.4007/annals.2005.161.679. |
[8] |
A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. Google Scholar |
[9] |
A. Ghosh and D. Kelmer,
A quantitative Oppenheim theorem for generic ternary quadratic forms, J. Mod. Dyn., 12 (2018), 1-12.
doi: 10.3934/jmd.2018001. |
[10] |
G. Margulis,
Formes quadratiques indéfinies et flots unipotents sur les spaces homogénes, C. R. Acad. Sci. Paris Ser. I, 304 (1987), 249-253.
|
[11] |
A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727. Google Scholar |
[12] |
C. A. Rogers,
The number of lattice points in a set, Proc. London Math. Soc., 6 (1956), 305-320.
|
[13] |
W. Schmidt,
A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529.
doi: 10.1090/S0002-9947-1960-0117222-9. |
[14] |
C. L. Siegel,
A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347.
doi: 10.2307/1969027. |
[15] |
J. M. VanderKam,
Values at integers of homogeneous polynomials, Duke Math. J., 97 (1999), 379-412.
doi: 10.1215/S0012-7094-99-09716-8. |
show all references
References:
[1] |
J. S. Athreya,
Random affine lattices, Contemp. Math., 639 (2015), 160-174.
|
[2] |
J. S. Athreya and G. A. Margulis,
Logarithm laws for unipotent flows. I, J. Mod. Dyn., 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
[3] |
J. N. Bernstein,
Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen., 6 (1972), 26-40.
|
[4] |
J. Bourgain,
A quantitative Oppenheim theorem for generic diagonal quadratic forms, Israel J. Math., 215 (2016), 503-512.
doi: 10.1007/s11856-016-1385-7. |
[5] |
A. Chambert-Loir and Y. Tschinkel,
Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math., 2 (2010), 351-429.
doi: 10.1142/S1793744210000223. |
[6] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.
doi: 10.2307/120984. |
[7] |
A. Eskin, G. Margulis and S. Mozes,
Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2-tori, Ann. of Math., 161 (2005), 679-725.
doi: 10.4007/annals.2005.161.679. |
[8] |
A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. Google Scholar |
[9] |
A. Ghosh and D. Kelmer,
A quantitative Oppenheim theorem for generic ternary quadratic forms, J. Mod. Dyn., 12 (2018), 1-12.
doi: 10.3934/jmd.2018001. |
[10] |
G. Margulis,
Formes quadratiques indéfinies et flots unipotents sur les spaces homogénes, C. R. Acad. Sci. Paris Ser. I, 304 (1987), 249-253.
|
[11] |
A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727. Google Scholar |
[12] |
C. A. Rogers,
The number of lattice points in a set, Proc. London Math. Soc., 6 (1956), 305-320.
|
[13] |
W. Schmidt,
A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529.
doi: 10.1090/S0002-9947-1960-0117222-9. |
[14] |
C. L. Siegel,
A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347.
doi: 10.2307/1969027. |
[15] |
J. M. VanderKam,
Values at integers of homogeneous polynomials, Duke Math. J., 97 (1999), 379-412.
doi: 10.1215/S0012-7094-99-09716-8. |
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