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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. 

[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.

[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. 

[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.

[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. 

[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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