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  May 2021 Early access  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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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. 

[5]

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

[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.

[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. 

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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. 

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[25]

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

[26]

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

[27]

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

[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.

[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.

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.

[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.

[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.

[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. 

[5]

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

[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.

[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. 

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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. 

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[25]

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

[26]

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

[27]

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

[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.

[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.

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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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