A quantitative Oppenheim theorem for generic ternary quadratic forms

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January 2018, 12: 1-8. doi: 10.3934/jmd.2018001

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

Received February 27, 2017 Revised November 10, 2017 Published December 2017

Fund Project: AG: Partially supported by ISF-UGC. DK: Partially supported by NSF grant DMS-1401747

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 [3].

Keywords: Values of quadratic forms, Diophantine approximation, flows on homogeneous spaces.

Mathematics Subject Classification: Primary: 11E20; Secondary: 37A17.

Citation: Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001

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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