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

A quantitative Oppenheim theorem for generic ternary quadratic forms

Anish Ghosh 1, and  Dubi Kelmer 2,

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.  Google Scholar

[2]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn. , to appear. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. Google Scholar

[8]

A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.  doi: 10.1112/blms.12023.  Google Scholar

[9]

A. Gorodnik and A. Nevo, The Ergodic Theory of Lattice Subgroups, Annals of Mathematics Studies, 172, Princeton University Press, Princeton, NJ, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. A. Rogers, Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.  doi: 10.1007/BF02392493.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn. , to appear. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, arXiv:1801.01027, 2018. Google Scholar

[8]

A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.  doi: 10.1112/blms.12023.  Google Scholar

[9]

A. Gorodnik and A. Nevo, The Ergodic Theory of Lattice Subgroups, Annals of Mathematics Studies, 172, Princeton University Press, Princeton, NJ, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. A. Rogers, Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.  doi: 10.1007/BF02392493.  Google Scholar

[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.  Google Scholar

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