May  2021, 41(5): 2205-2225. doi: 10.3934/dcds.2020359

Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

Research Institute of Mathematics, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul, 08826, South Korea

Received  January 2020 Revised  August 2020 Published  October 2020

Fund Project: This paper is supported by the Samsung Science and Technology Foundation under project No. SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) grant funded by the Korea government under project No. 0409-20200150

The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms $ q $ of rank at least 5, the number of integral vectors $ \mathbf v $ such that $ q( \mathbf v) $ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $ \mathbf v $ such that $ q( \mathbf v) $ is in the same interval.

In rank $ 3 $ or $ 4 $, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms([8, Theorem 2.4]). In this paper, we extend this result to the $ S $-arithmetic version.

Citation: Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359
References:
[1]

P. Abramenko and K. S. Brown, Buildings. Theory and Applications, Graduate Texts in Mathematics, 248. Springer, New York, 2008. doi: 10.1007/978-0-387-78835-7.  Google Scholar

[2]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn., 12 (2018), 9-16.  doi: 10.3934/jmd.2018002.  Google Scholar

[3]

P. BandiA. Ghosh and J. Han, A generic effective Oppenheim theorem for systems of forms, J. Number Thoery, 218 (2020), 311-333.  doi: 10.1016/j.jnt.2020.07.002.  Google Scholar

[4]

Y. Benoist, Five lectures on lattices in semisimple Lie groups, Géométries à Courbure Négative ou Nulle, Groupes Discrets et Rigidités, Sémin. Congr., Soc. Math. France, Paris, 18 (2009), 117-176.   Google Scholar

[5]

A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compos. Math., 83 (1992), 347-372.   Google Scholar

[6]

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

[7]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[8]

A. EskinG. 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

[9]

A. EskinG. 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

[10]

A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, preprint, arXiv: 1801.01027. Google Scholar

[11]

A. Ghosh and D. Kelmer, A quantitative Oppenheim theorem for generic ternary quadratic forms, J. Mod. Dyn., 12 (2018), 1-8.  doi: 10.3934/jmd.2018001.  Google Scholar

[12]

A. Gorodnik, Oppenheim conjecture for pairs consisting of a linear form and a quadratic form, Trans. Amer. Math. Soc., 356 (2004), 4447-4463.  doi: 10.1090/S0002-9947-04-03473-7.  Google Scholar

[13]

J. HanS. Lim and K. Mallahi-Karai, Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points, J. Mod. Dyn., 11 (2017), 501-550.  doi: 10.3934/jmd.2017020.  Google Scholar

[14]

D. Kelmer and S. Yu, Values of random polynomials in shrinking targets, preprint, arXiv: 1812.04541. Google Scholar

[15]

D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581.  doi: 10.4171/CMH/102.  Google Scholar

[16]

Y. Lazar, Values of pairs involving one quadratic form and one linear form at $S$-integral points, J. Number Theory, 181 (2017), 200-217.  doi: 10.1016/j.jnt.2017.06.003.  Google Scholar

[17]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253.   Google Scholar

[18]

H. Oh, Uniform pointwise bounds for matrix coefficients, Duke Math. J., 113 (2002), 133-192.   Google Scholar

[19]

A. Oppenheim, The Minima of Indefinite Quaternary Quadratic Forms, Thesis (Ph.D.)–The University of Chicago, 1930.  Google Scholar

[20]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139. Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[21]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382.  doi: 10.1215/S0012-7094-95-07710-2.  Google Scholar

[22]

G. Robertson, Euclidean Buildings, (lecture), "Arithmetic Geometry and Noncommutative Geometry", Masterclass, Utrecht, 2010. Google Scholar

[23]

O. Sargent, Density of values of linear maps on quadratic surfaces, J. Number Theory, 143 (2014), 363-384.  doi: 10.1016/j.jnt.2014.04.020.  Google Scholar

[24]

O. Sargent, Equidistribution of values of linear forms on quadratic surfaces, Algebra Number Theory, 8 (2014), 895-932.  doi: 10.2140/ant.2014.8.895.  Google Scholar

[25]

W. M. Schmidt, Approximation to algebraic numbers, Enseignement Math., 17 (1971), 187-253.   Google Scholar

[26]

J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[27]

J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[28]

T. A. Springer, Linear Algebraic Groups, Second edition, Progress in Mathematics, 9, Birkh user Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[29]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 26 (2000), 265-297.  doi: 10.2969/aspm/02610265.  Google Scholar

show all references

References:
[1]

P. Abramenko and K. S. Brown, Buildings. Theory and Applications, Graduate Texts in Mathematics, 248. Springer, New York, 2008. doi: 10.1007/978-0-387-78835-7.  Google Scholar

[2]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn., 12 (2018), 9-16.  doi: 10.3934/jmd.2018002.  Google Scholar

[3]

P. BandiA. Ghosh and J. Han, A generic effective Oppenheim theorem for systems of forms, J. Number Thoery, 218 (2020), 311-333.  doi: 10.1016/j.jnt.2020.07.002.  Google Scholar

[4]

Y. Benoist, Five lectures on lattices in semisimple Lie groups, Géométries à Courbure Négative ou Nulle, Groupes Discrets et Rigidités, Sémin. Congr., Soc. Math. France, Paris, 18 (2009), 117-176.   Google Scholar

[5]

A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compos. Math., 83 (1992), 347-372.   Google Scholar

[6]

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

[7]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[8]

A. EskinG. 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

[9]

A. EskinG. 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

[10]

A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, preprint, arXiv: 1801.01027. Google Scholar

[11]

A. Ghosh and D. Kelmer, A quantitative Oppenheim theorem for generic ternary quadratic forms, J. Mod. Dyn., 12 (2018), 1-8.  doi: 10.3934/jmd.2018001.  Google Scholar

[12]

A. Gorodnik, Oppenheim conjecture for pairs consisting of a linear form and a quadratic form, Trans. Amer. Math. Soc., 356 (2004), 4447-4463.  doi: 10.1090/S0002-9947-04-03473-7.  Google Scholar

[13]

J. HanS. Lim and K. Mallahi-Karai, Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points, J. Mod. Dyn., 11 (2017), 501-550.  doi: 10.3934/jmd.2017020.  Google Scholar

[14]

D. Kelmer and S. Yu, Values of random polynomials in shrinking targets, preprint, arXiv: 1812.04541. Google Scholar

[15]

D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581.  doi: 10.4171/CMH/102.  Google Scholar

[16]

Y. Lazar, Values of pairs involving one quadratic form and one linear form at $S$-integral points, J. Number Theory, 181 (2017), 200-217.  doi: 10.1016/j.jnt.2017.06.003.  Google Scholar

[17]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253.   Google Scholar

[18]

H. Oh, Uniform pointwise bounds for matrix coefficients, Duke Math. J., 113 (2002), 133-192.   Google Scholar

[19]

A. Oppenheim, The Minima of Indefinite Quaternary Quadratic Forms, Thesis (Ph.D.)–The University of Chicago, 1930.  Google Scholar

[20]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139. Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[21]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382.  doi: 10.1215/S0012-7094-95-07710-2.  Google Scholar

[22]

G. Robertson, Euclidean Buildings, (lecture), "Arithmetic Geometry and Noncommutative Geometry", Masterclass, Utrecht, 2010. Google Scholar

[23]

O. Sargent, Density of values of linear maps on quadratic surfaces, J. Number Theory, 143 (2014), 363-384.  doi: 10.1016/j.jnt.2014.04.020.  Google Scholar

[24]

O. Sargent, Equidistribution of values of linear forms on quadratic surfaces, Algebra Number Theory, 8 (2014), 895-932.  doi: 10.2140/ant.2014.8.895.  Google Scholar

[25]

W. M. Schmidt, Approximation to algebraic numbers, Enseignement Math., 17 (1971), 187-253.   Google Scholar

[26]

J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[27]

J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[28]

T. A. Springer, Linear Algebraic Groups, Second edition, Progress in Mathematics, 9, Birkh user Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[29]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 26 (2000), 265-297.  doi: 10.2969/aspm/02610265.  Google Scholar

Figure 1.  The 3-dimensional hyperbolic space $ {\mathbb{H}}^3 $. The measure of the set in (4) is equal to the Lebesgue measure of the grey area on the top of the sphere
Figure 2.  Apartment $ \mathcal A_0 $ of $ \mathcal{B}_3 $. $ K_p\setminus {\mathrm{SO}}(2x_1x_3-x_2^2) $ is embedded in the inverse image of the blue line
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