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A quantitative Oppenheim theorem for generic ternary quadratic forms
| 1. | School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India |
| 2. | Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA |
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain [
References:
| [1] |
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. |
| [2] |
J. S. Athreya and G. A. Margulis,
Values of random polynomials at integer points,
J. Mod. Dyn. , to appear. |
| [3] |
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. |
| [4] |
S. G. Dani and G. A. Margulis,
Limit distributions of orbits of unipotent flows and values of quadratic forms, Adv. in Soviet Math., 16 (1993), 91-137.
|
| [5] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
| [6] |
A. Ghosh, A. Gorodnik and A. Nevo,
Best possible rates of distribution of dense lattice orbits in homogeneous spaces,
J. Reine Angew. Math. , to appear. |
| [7] |
A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. |
| [8] |
A. Ghosh and D. Kelmer,
Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.
doi: 10.1112/blms.12023. |
| [9] |
A. Gorodnik and A. Nevo,
The Ergodic Theory of Lattice Subgroups,
Annals of Mathematics Studies, 172, Princeton University Press, Princeton, NJ, 2010. |
| [10] |
E. Lindenstrauss and G. Margulis,
Effective estimates on indefinite ternary forms, Israel J. Math., 203 (2014), 445-499.
doi: 10.1007/s11856-014-1110-3. |
| [11] |
G. A. Margulis,
Discrete subgroups and ergodic theory,
in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989,377-398. |
| [12] |
H. Oh,
Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France, 126 (1998), 355-380.
doi: 10.24033/bsmf.2329. |
| [13] |
C. A. Rogers,
Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.
doi: 10.1007/BF02392493. |
| [14] |
P. Sarnak,
Values at integers of binary quadratic forms,
in Harmonic Analysis and Number Theory (Montreal, PQ, 1996), CMS Conf. Proc., 21, AMS, Providence, RI, (1997), 181-203. |
show all references
References:
| [1] |
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. |
| [2] |
J. S. Athreya and G. A. Margulis,
Values of random polynomials at integer points,
J. Mod. Dyn. , to appear. |
| [3] |
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. |
| [4] |
S. G. Dani and G. A. Margulis,
Limit distributions of orbits of unipotent flows and values of quadratic forms, Adv. in Soviet Math., 16 (1993), 91-137.
|
| [5] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
| [6] |
A. Ghosh, A. Gorodnik and A. Nevo,
Best possible rates of distribution of dense lattice orbits in homogeneous spaces,
J. Reine Angew. Math. , to appear. |
| [7] |
A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. |
| [8] |
A. Ghosh and D. Kelmer,
Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.
doi: 10.1112/blms.12023. |
| [9] |
A. Gorodnik and A. Nevo,
The Ergodic Theory of Lattice Subgroups,
Annals of Mathematics Studies, 172, Princeton University Press, Princeton, NJ, 2010. |
| [10] |
E. Lindenstrauss and G. Margulis,
Effective estimates on indefinite ternary forms, Israel J. Math., 203 (2014), 445-499.
doi: 10.1007/s11856-014-1110-3. |
| [11] |
G. A. Margulis,
Discrete subgroups and ergodic theory,
in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989,377-398. |
| [12] |
H. Oh,
Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France, 126 (1998), 355-380.
doi: 10.24033/bsmf.2329. |
| [13] |
C. A. Rogers,
Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.
doi: 10.1007/BF02392493. |
| [14] |
P. Sarnak,
Values at integers of binary quadratic forms,
in Harmonic Analysis and Number Theory (Montreal, PQ, 1996), CMS Conf. Proc., 21, AMS, Providence, RI, (1997), 181-203. |
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