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  2018, 12: 9-16. doi: 10.3934/jmd.2018002

Values of random polynomials at integer points

Jayadev S. Athreya 1, and  Gregory A. Margulis 2,

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

Received  April 12, 2017 Revised  November 10, 2017 Published  February 2018

Fund Project: This material is based upon work while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2015 semester, supported by the National Science Foundation Grant DMS 0932078 000.
JSA: Partially supported by NSF CAREER grant DMS 1559860, NSF grant DMS 1069153, and grants DMS 1107452,1107263,1107367 "RNMS: GEometric structures And Representation varieties (the GEAR Network)".
GAM: Supported by NSF grant DMS 1265695.

Using classical results of Rogers [12, Theorem 1] bounding the L2-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

Keywords: Random lattices, Rogers' formula, Oppenheim's conjecture.
Mathematics Subject Classification: Primary: 11H60, 11P21, 11C08.
Citation: Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
References:
[1]

J. S. Athreya, Random affine lattices, Contemp. Math., 639 (2015), 160-174.   Google Scholar

[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.  Google Scholar

[3]

J. N. Bernstein, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen., 6 (1972), 26-40.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Margulis, Formes quadratiques indéfinies et flots unipotents sur les spaces homogénes, C. R. Acad. Sci. Paris Ser. I, 304 (1987), 249-253.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347.  doi: 10.2307/1969027.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. S. Athreya, Random affine lattices, Contemp. Math., 639 (2015), 160-174.   Google Scholar

[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.  Google Scholar

[3]

J. N. Bernstein, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen., 6 (1972), 26-40.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Margulis, Formes quadratiques indéfinies et flots unipotents sur les spaces homogénes, C. R. Acad. Sci. Paris Ser. I, 304 (1987), 249-253.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347.  doi: 10.2307/1969027.  Google Scholar

[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.  Google Scholar

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