A quantitative Oppenheim theorem for generic ternary quadratic forms

Anish Ghosh; Dubi Kelmer

doi:10.3934/jmd.2018001

Article Contents

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

This volume Previous Article Next Article Roy Adler's publications and patents

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

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.