American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations
  • DCDS Home
  • This Issue
  • Next Article
    Liouville type theorems for fractional and higher-order fractional systems
doi: 10.3934/dcds.2020359

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

Jiyoung Han

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.

Keywords: Quantitative Oppenheim conjecture, $ S $-arithmetic space, action of quadratic form-preserving groups, low-dimensional $ p $-adic symmetric space, Euclidean building.
Mathematics Subject Classification: Primary: 22F30, 22E45, 20F65; Secondary: 20E08, 37P55.
Citation: Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems - A, 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. Bandi, A. 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. 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

[9]

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

[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. Han, S. 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. Bandi, A. 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. 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

[9]

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

[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. Han, S. 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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[3]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[4]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[5]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018
[6]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[7]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[8]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[9]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015
[10]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[11]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[12]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379
[13]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[14]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[15]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076
[16]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[17]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[18]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270
[19]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349
[20]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

2019 Impact Factor: 1.338

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences