
-
Previous Article
A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"
- DCDS Home
- This Issue
-
Next Article
An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth
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 |
The celebrated result of Eskin, Margulis and Mozes [
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([
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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[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. |


[1] |
Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230 |
[2] |
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 |
[3] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
[4] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
[5] |
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 |
[6] |
Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048 |
[7] |
Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 |
[8] |
Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 |
[9] |
Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169 |
[10] |
Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020123 |
[11] |
Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 |
[12] |
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 |
[13] |
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 |
[14] |
Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065 |
[15] |
Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020374 |
[16] |
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 |
[17] |
Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020122 |
[18] |
Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383 |
[19] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
[20] |
Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]