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A special case of effective equidistribution with explicit constants

Published online by Cambridge University Press:  14 March 2011

A. MOHAMMADI
A. MOHAMMADI*
Affiliation:
Mathematics Department, University of Chicago, Chicago, IL, USA (email: [email protected])
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Abstract

An effective equidistribution with explicit constants for the isometry group of rational forms with signature (2,1) is proved. As an application we get an effective discreteness of the Markov spectrum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 237 - 247
Copyright
Copyright © Cambridge University Press 2011

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References

[BR02]Bernstein, J. and Reznikov, A.. Sobolev norms of automorphic functionals. Int. Math. Res. Not. (40) (2002), 2155–2174.CrossRefGoogle Scholar
[CaS55]Cassels, J. W. and Swinnerton-Dyer, H. P.. On the product of three homogeneous linear forms and indefinite ternary quadratic forms. Philos. Trans. R. Soc. Lond. 248(940) (1955), 73–96.Google Scholar
[Co01]Cogdell, J. W.. On sums of three squares. Les XXIIémes Journées Arithmetiques (Lille, 2001). J. Théor. Nombres Bordeaux 15(1) (2003), 33–44.CrossRefGoogle Scholar
[DM89]Dani, S. G. and Margulis, G. A.. Values of quadratic forms at primitive integral points. Invent. Math. 98(2) (1989), 405–424.CrossRefGoogle Scholar
[E06]Einsiedler, M.. Ratner’s theorem on SL2(ℝ)-invariant measures. Jahresber. Deutsch. Math.-Verein. 108(3) (2006), 143–164.Google Scholar
[EMV09]Einsiedler, M., Margulis, G. A. and Venkatesh, A.. Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. Invent. Math. 177(1) (2009), 137–212.CrossRefGoogle Scholar
[KS03]Kim, H. H. and Sarnak, P.. Refined estimates towards the Ramanujan and Selberg conjectures, Appendix to H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. J. Amer. Math. Soc. 16(1) (2003), 139–183.CrossRefGoogle Scholar
[KM98]Kleinbock, D. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), 339–360.CrossRefGoogle Scholar
[Mar86]Margulis, G. A.. Indefinite quadratic forms and unipotent flows on homogeneous spaces. Proc. Semester on Dynamical Systems and Ergodic Theory (Warsaw 1986) (Banach Center Publications, 23). PWN, Warsaw, 1989, pp. 399–409.Google Scholar
[Mar90a]Margulis, G. A.. Orbits of group actions and values of quadrtic forms at integral points. Festschrift in Honour of I. I. Piatetski-Shapiro (Israel Mathematical Conference Proceedings, 3). Weizmann Science Press, Israel, Jerusalem, 1990, pp. 127–151.Google Scholar
[Mar91]Margulis, G. A.. Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory. Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vol. I, II. Mathematical Society of Japan, Tokyo, 1991, pp. 193–215.Google Scholar
[MT94]Margulis, G. A. and Tomanov, G.. Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116(1–3) (1994), 347–392.CrossRefGoogle Scholar
[Mark]Markov, A. A.. On binary quadratic forms of positive determinant. SPb, 1880. Usp. Mat. Nauk (3) 5 (1948), 7–51.Google Scholar
[MS95]Mozes, S. and Shah, N.. On the space of ergodic invariant measures of unipotent flows. Ergod. Th. & Dynam. Sys. 15(1) (1995), 149–159.CrossRefGoogle Scholar
[Oh02]Oh, H.. Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants. Duke Math. J. 113 (2002), 133–192.CrossRefGoogle Scholar
[R90]Ratner, M.. On measure rigidity of unipotent subgroups of semi-simple groups. Acta Math. 165 (1990), 229–309.CrossRefGoogle Scholar
[R91]Rather, M.. Raghunathan topological conjecture and distributions of unipotent flows. Duke Math. J. 63 (1991), 235–280.Google Scholar
[R92]Ratner, M.. On Raghunathan’s measure conjecture. Ann. of Math. (2) 134 (1992), 545–607.CrossRefGoogle Scholar
[R95]Ratner, M.. Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups. Duke Math. J. 77(2) (1995), 275–382.CrossRefGoogle Scholar