Comptes Rendus
Dynamical systems
Polynomial effective equidistribution
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 507-520.

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of SL 2 (𝔩) in arithmetic quotients of SL 2 () and SL 2 (𝔩)×SL 2 (𝔩).

The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

Nous prouvons des théorémes d’équidistribution effectifs, avec un taux d’erreur polynomial pour les orbites des sous-groupes unipotents de SL 2 (𝔩) en quotients arithmétiques de SL 2 () et SL 2 (𝔩)×SL 2 (𝔩).

La preuve est basée sur l’utilisation d’une fonction de Margulis, des outils de la géométrie d’incidence, et le trou spectral de l’espace ambiant.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.411

Elon Lindenstrauss 1; Amir Mohammadi 2; Zhiren Wang 3

1 The Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
2 Department of Mathematics, University of California, San Diego, CA 92093, USA
3 Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Elon Lindenstrauss; Amir Mohammadi; Zhiren Wang. Polynomial effective equidistribution. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 507-520. doi : 10.5802/crmath.411. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.411/

[1] Jean Bourgain; Alex Furman; Elon Lindenstrauss; Shahar Mozes Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, J. Am. Math. Soc., Volume 24 (2011) no. 1, pp. 231-280 | DOI | MR | Zbl

[2] Jon Chaika; John Smillie; Barak Weiss Tremors and horocycle dynamics on the moduli space of translation surfaces (2020) (https://arxiv.org/abs/2004.04027)

[3] Sam Chow; Lei Yang An effective Ratner equidistribution theorem for multiplicative Diophantine approximation on planar lines (2019) (https://arxiv.org/abs/1902.06081)

[4] Laurent Clozel; Hee Oh; Emmanuel Ullmo Hecke operators and equidistribution of Hecke points, Invent. Math., Volume 144 (2001) no. 2, pp. 327-351 | DOI | MR | Zbl

[5] Shrikrishna G. Dani On orbits of unipotent flows on homogeneous spaces, Ergodic Theory Dyn. Syst., Volume 4 (1984) no. 1, pp. 25-34 | DOI | MR | Zbl

[6] Shrikrishna G. Dani On orbits of unipotent flows on homogeneous spaces. II, Ergodic Theory Dyn. Syst., Volume 6 (1986) no. 2, pp. 167-182 | DOI | MR | Zbl

[7] Shrikrishna G. Dani; Gregory Margulis Values of quadratic forms at primitive integral points, Invent. Math., Volume 98 (1989) no. 2, pp. 405-424 | DOI | MR | Zbl

[8] Shrikrishna G. Dani; Gregory Margulis Orbit closures of generic unipotent flows on homogeneous spaces of SL(3,R), Math. Ann., Volume 286 (1990) no. 1-3, pp. 101-128 | DOI | MR | Zbl

[9] Shrikrishna G. Dani; Gregory Margulis Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci., Math. Sci., Volume 101 (1991) no. 1, pp. 1-17 | DOI | MR | Zbl

[10] Manfred Einsiedler; Gregory Margulis; Amir Mohammadi; Akshay Venkatesh Effective equidistribution and property (τ), J. Am. Math. Soc., Volume 33 (2020) no. 1, pp. 223-289 | DOI | MR | Zbl

[11] Manfred Einsiedler; Gregory Margulis; Akshay Venkatesh Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., Volume 177 (2009) no. 1, pp. 137-212 | DOI | MR | Zbl

[12] Alex Eskin; Gregory Margulis; Shahar Mozes Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. Math., Volume 147 (1998) no. 1, pp. 93-141 | DOI | MR | Zbl

[13] Alex Eskin; Maryam Mirzakhani Invariant and stationary measures for the SL(2,) action on moduli space, Publ. Math., Inst. Hautes Étud. Sci., Volume 127 (2018), pp. 95-324 | DOI | MR | Zbl

[14] Alex Eskin; Maryam Mirzakhani; Amir Mohammadi Isolation, equidistribution, and orbit closures for the SL(2,) action on moduli space, Ann. Math., Volume 182 (2015) no. 2, pp. 673-721 | DOI | MR | Zbl

[15] Alex Eskin; Shahar Mozes Margulis Functions and Their Applications, Dynamics, Geometry, Number Theory: the impact of Margulis on modern mathematics (David Fisher; Dmitry Kleinbock; Gregory Soifer, eds.), University of Chicago Press, 2022 | DOI | Zbl

[16] Livio Flaminio; Giovanni Forni; James Tanis Effective equidistribution of twisted horocycle flows and horocycle maps, Geom. Funct. Anal., Volume 26 (2016) no. 5, pp. 1359-1448 | DOI | MR | Zbl

[17] Giovanni Forni Limits of geodesic push-forwards of horocycle invariant measures, Ergodic Theory Dyn. Syst., Volume 41 (2021) no. 9, pp. 2782-2804 | DOI | MR | Zbl

[18] Howard Garland; Madabusi S. Raghunathan Fundamental Domains for Lattices in -rank 1 Semisimple Lie Groups, Ann. Math., Volume 92 (1970) no. 2, pp. 279-326 | DOI | Zbl

[19] Ben Green; Terence Tao The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. Math., Volume 175 (2012) no. 2, pp. 465-540 | DOI | MR | Zbl

[20] Weikun He; Nicolas de Saxcé Linear random walks on the torus (2019) (https://arxiv.org/abs/1910.13421v1)

[21] H. Jacquet; Robert P. Langlands Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, Springer, 1970 | DOI | MR | Zbl

[22] Antti Käenmäki; Tuomas Orponen; Laura Venieri A Marstrand-type restricted projection theorem in 3 (2017) (https://arxiv.org/abs/1708.04859v1)

[23] Asaf Katz Quantitative disjointness of nilflows from horospherical flows (2019) (https://arxiv.org/abs/1910.04675)

[24] Wooyeon Kim Effective equidistribution of expanding translates in the space of affine lattices (2021) (https://arxiv.org/abs/2110.00706)

[25] Dmitry Y. Kleinbock; Gregory Margulis Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinai’s Moscow Seminar on Dynamical Systems (American Mathematical Society Translations, Series 2), Volume 171, American Mathematical Society, 1996, pp. 141-172 | Zbl

[26] Elon Lindenstrauss; Gregory Margulis Effective estimates on indefinite ternary forms, Isr. J. Math., Volume 203 (2014) no. 1, pp. 445-499 | DOI | MR | Zbl

[27] Elon Lindenstrauss; Amir Mohammadi Polynomial effective density in quotients of 3 and 2 × 2 (2021) (https://arxiv.org/abs/2112.14562v1)

[28] Elon Lindenstrauss; Amir Mohammadi; Gregory Margulis; Nimish A. Shah Quantitative behavior of unipotent flows and an effective avoidance principle (2019) (https://arxiv.org/abs/1904.00290v1)

[29] Gregory Margulis The action of unipotent groups in a lattice space, Mat. Sb., N. Ser., Volume 86(128) (1971), pp. 552-556 | MR | Zbl

[30] Gregory Margulis Indefinite quadratic forms and unipotent flows on homogeneous spaces, Dynamical systems and ergodic theory (Warsaw, 1986) (Banach Center Publications), Volume 23, Polish Academy of Sciences, 1989, pp. 399-409 | Zbl

[31] Gregory Margulis Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 17, Springer, 1991 | DOI | MR | Zbl

[32] Amir Mohammadi; Hee Oh Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifold (2020) (https://arxiv.org/abs/2002.06579v1)

[33] Marina Ratner On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., Volume 165 (1990) no. 3-4, pp. 229-309 | DOI | Zbl

[34] Marina Ratner On Raghunathan’s measure conjecture, Ann. Math., Volume 134 (1991) no. 3, pp. 545-607 | DOI | Zbl

[35] Marina Ratner Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J., Volume 63 (1991) no. 1, pp. 235-280 | DOI | Zbl

[36] Wilhelm Schlag On continuum incidence problems related to harmonic analysis, J. Funct. Anal., Volume 201 (2003) no. 2, pp. 480-521 | DOI | Zbl

[37] Atle Selberg On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, 1960, pp. 147-164 | MR | Zbl

[38] Atle Selberg On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, American Mathematical Society, 1965, pp. 1-15 | MR | Zbl

[39] Nimish A. Shah Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., Volume 289 (1991) no. 2, pp. 315-334 | DOI | MR | Zbl

[40] Nimish A. Shah Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci., Math. Sci., Volume 106 (1996) no. 2, pp. 105-125 | DOI | MR | Zbl

[41] Andreas Strömbergsson An effective Ratner equidistribution result for SL(2,) 2 , Duke Math. J., Volume 164 (2015) no. 5, pp. 843-902 | DOI | MR | Zbl

[42] Akshay Venkatesh Sparse equidistribution problems, period bounds and subconvexity, Ann. Math., Volume 172 (2010) no. 2, pp. 989-1094 | DOI | MR | Zbl

[43] André Weil On Discrete Subgroups of Lie Groups, Ann. Math., Volume 72 (1960) no. 2, pp. 369-384 | DOI | Zbl

[44] André Weil Remarks on the Cohomology of Groups, Ann. Math., Volume 80 (1964) no. 1, pp. 149-157 | DOI | Zbl

[45] Thomas H. Wolff Local smoothing type estimates on L p for large p, Geom. Funct. Anal., Volume 10 (2000) no. 5, pp. 1237-1288 | DOI | MR | Zbl

[46] Joshua Zahl L 3 estimates for an algebraic variable coefficient Wolff circular maximal function, Rev. Mat. Iberoam., Volume 28 (2012) no. 4, pp. 1061-1090 | DOI | MR | Zbl

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