On discrepancy, intrinsic Diophantine approximation, and spectral gaps

Alexander Gorodnik; Amos Nevo

doi:10.5802/jtnb.1275

Alexander Gorodnik ¹ ; Amos Nevo ^2,³

¹ Institute für Mathematik, Universität Zürich Switzerland
² Department of Mathematics, Technion, Israel
³ Department of Mathematics, University of Chicago, USA

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 127-174.

Résumé
Abstract

Dans le présent article, nous établissons des bornes pour la taille de l’écart spectral pour les actions de groupe sur les espaces homogènes. Notre approche est basée sur l’estimation des normes des opérateurs de moyennage appropriés, et nous développons des techniques pour établir des bornes supérieures et inférieures pour de telles normes. Nous montrerons que ce problème analytique est étroitement lié au problème arithmétique de l’établissement de bornes sur la divergence de distribution pour les points rationnels sur les variétés de groupes algébriques. Comme application, nous montrons comment établir une borne effective pour la propriété $(τ)$ des sous-groupes de congruence des treillis arithmétiques dans les groupes algébriques qui sont des formes de ${SL}_{2}$ , en utilisant des estimations dans l’approximation diophantienne intrinsèque qui découlent de l’analyse de Heath-Brown des points rationnels sur des variétés quadratiques de dimension $3$ .

In the present paper we establish bounds for the size of the spectral gap for group actions on homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $(τ)$ of congruence subgroups of arithmetic lattices in algebraic groups which are forms of ${SL}_{2}$ , using estimates in intrinsic Diophantine approximation which follow from Heath-Brown’s analysis of rational points on $3$ -dimensional quadratic surfaces.

Reçu le : 2022-02-28
Révisé le : 2023-08-24
Accepté le : 2023-09-15
Publié le : 2024-07-25

DOI : 10.5802/jtnb.1275

Classification : 22E46, 11F70, 11F72
Mots clés : Semisimple group, discrepancy, spectral gap, Diophantine approximation, automorphic representation

Affiliations des auteurs :

Alexander Gorodnik ¹ ; Amos Nevo ^{2, 3}

¹ Institute für Mathematik, Universität Zürich Switzerland
² Department of Mathematics, Technion, Israel
³ Department of Mathematics, University of Chicago, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2024__36_1_127_0,
     author = {Alexander Gorodnik and Amos Nevo},
     title = {On discrepancy, intrinsic {Diophantine} approximation, and spectral gaps},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {127--174},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {1},
     year = {2024},
     doi = {10.5802/jtnb.1275},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1275/}
}

TY  - JOUR
AU  - Alexander Gorodnik
AU  - Amos Nevo
TI  - On discrepancy, intrinsic Diophantine approximation, and spectral gaps
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2024
SP  - 127
EP  - 174
VL  - 36
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1275/
DO  - 10.5802/jtnb.1275
LA  - en
ID  - JTNB_2024__36_1_127_0
ER  -

%0 Journal Article
%A Alexander Gorodnik
%A Amos Nevo
%T On discrepancy, intrinsic Diophantine approximation, and spectral gaps
%J Journal de théorie des nombres de Bordeaux
%D 2024
%P 127-174
%V 36
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1275/
%R 10.5802/jtnb.1275
%G en
%F JTNB_2024__36_1_127_0

Alexander Gorodnik; Amos Nevo. On discrepancy, intrinsic Diophantine approximation, and spectral gaps. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 127-174. doi : 10.5802/jtnb.1275. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1275/

Bibliographie
Cité par

[1] Claire Anantharaman-Delaroche On spectral characterizations of amenability, Isr. J. Math., Volume 137 (2003), pp. 1-33 | DOI | Zbl

[2] Bachir Bekka; Pierre de la Harpe; Alain Valette Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiii+472 pages | DOI

[3] Valentin Blomer; Farrell Brumley On the Ramanujan conjecture over number fields, Ann. Math., Volume 174 (2011) no. 1, pp. 581-605 | DOI | Zbl

[4] Valentin Blomer; Farrell Brumley The role of the Ramanujan conjecture in analytic number theory, Bull. Am. Math. Soc., Volume 50 (2013) no. 2, pp. 267-320 | DOI | Zbl

[5] Marc Burger; Jian-Shu Li; Peter Sarnak Ramanujan duals and automorphic spectrum, Bull. Am. Math. Soc., Volume 26 (1992) no. 2, pp. 253-257 | DOI | Zbl

[6] Marc Burger; Peter Sarnak Ramanujan duals. II, Invent. Math., Volume 106 (1991) no. 1, pp. 1-11 | DOI | Zbl

[7] Pierre Cartier Representations of $𝔭$ -adic groups: a survey, Automorphic forms, representations and $L$ -functions (Oregon State Univ., Corvallis, 1977) (Proceedings of Symposia in Pure Mathematics), Volume 33.1, American Mathematical Society, 1979, pp. 111-155 | DOI | Zbl

[8] Laurent Clozel Démonstration de la conjecture $τ$ , Invent. Math., Volume 151 (2003) no. 2, pp. 297-328 | DOI | Zbl

[9] Laurent Clozel Spectral theory of automorphic forms, Automorphic forms and applications (IAS/Park City Mathematics Series), Volume 12, American Mathematical Society, 2007, pp. 43-93 | Zbl

[10] Michael G. Cowling The Kunze–Stein phenomenon, Ann. Math. (1978), pp. 209-234 | DOI | Zbl

[11] Harold Davenport Analytic methods for Diophantine equations and Diophantine inequalities, Cambridge Mathematical Library, Cambridge University Press, 2005, xx+140 pages | DOI

[12] William Duke; John Friedlander; Henryk Iwaniec Bounds for automorphic L-functions, Invent. Math., Volume 112 (1993) no. 1, pp. 1-8 | DOI | Zbl

[13] Pierre Eymard Moyennes et représentations unitaires, Lecture Notes in Mathematics, 300, Springer, 1972, 113 pages | DOI

[14] Tobias Finis; Jasmin Matz On the asymptotics of Hecke operators for reductive groups, Math. Ann., Volume 380 (2021) no. 3-4, pp. 1037-1104 | DOI | Zbl

[15] Ramesh Gangolli; Veeravalli S. Varadarajan Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 101, Springer, 1988, xiv+365 pages | DOI

[16] Israil’ M. Gel’fand; Mark I. Graev; Ilya I. Piatetski-Shapiro Representation theory and automorphic functions, W. B. Saunders Co., 1969

[17] Anish Ghosh; Alexander Gorodnik; Amos Nevo Counting intrinsic Diophantine approximations in simple algebraic groups, Isr. J. Math., Volume 251 (2022) no. 2, pp. 443-466 | DOI | Zbl

[18] Alexander Gorodnik; Amos Nevo The ergodic theory of lattice subgroups, Annals of Mathematics Studies, 172, Princeton University Press, 2009, 136 pages

[19] Alexander Gorodnik; Amos Nevo Quantitative ergodic theorems and their number theoretic applications, Bull. Am. Math. Soc., Volume 52 (2015) no. 1, pp. 65-113 | DOI | Zbl

[20] Alexander Gorodnik; Amos Nevo Discrepancy of rational points in simple algebraic groups (2021) (https://arxiv.org/abs/2104.06765)

[21] Frederik P. Greenleaf Amenable actions of locally compact groups, J. Funct. Anal., Volume 4 (1969), pp. 295-315 | DOI | Zbl

[22] D. Roger Heath-Brown A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math., Volume 481 (1996), pp. 149-206 | Zbl

[23] Alessandria Iozzi; Amos Nevo Algebraic hulls and the Følner property, Geom. Funct. Anal., Volume 6 (1996) no. 4, pp. 666-688 | DOI | Zbl

[24] David A. Kazhdan Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl., Volume 1 (1967), pp. 63-65 | DOI | Zbl

[25] Henry H. Kim; Peter Sarnak Refined estimates towards the Ramanujan and Selberg conjectures appendix in H. H. Kim, “Functoriality for the exterior square of $GL (4)$ and the symmetric fourth of $GL (2)$ ”, J. Am. Math. Soc. 16 (2003), no. 1, 139–183

[26] Robert P. Langlands On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, 544, Springer, 1976, v+337 pages | DOI

[27] Ian G. Macdonald Spherical functions on a $p$ -adic Chevalley group, Bull. Am. Math. Soc., Volume 74 (1968) no. 3, pp. 520-525 | DOI | Zbl

[28] Pertti Mattila Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, 1995, xii+343 pages | DOI

[29] Jasmin Matz; Nicolas Templier Sato-Tate equidistribution for families of Hecke-Maass forms on $SL (n, ℝ) / SO (n)$ , Algebra Number Theory, Volume 15 (2021) no. 6, pp. 1343-1428 | DOI | Zbl

[30] Amos Nevo Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett., Volume 5 (1998) no. 3, pp. 305-325 | DOI | Zbl

[31] Vladimir Platonov; Andrei Rapinchuk Algebraic groups and number theory, Pure and Applied Mathematics, 139, Academic Press Inc., 1994, xi+614 pages

[32] J. Rogawski Modular forms, the Ramanujan conjecture, and the Jacquet-Langlands correspondence (appendix in A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser, 1994)

[33] Peter Sarnak Notes on the generalized Ramanujan conjectures, Harmonic analysis, the trace formula, and Shimura varieties (Toronto, 2003) (Clay Mathematics Proceedings), Volume 4, American Mathematical Society, 2005, pp. 659-685 | Zbl

[34] Peter Sarnak; Xiaoxi Xue Bounds for multiplicities of automorphic representations, Duke Math. J., Volume 64 (1991) no. 1, pp. 207-227 | Zbl

[35] Atle Selberg Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., n. Ser., Volume 20 (1956), pp. 47-87 | Zbl

[36] Atle Selberg On estimation of Fourier coefficients of modular forms, Theory of numbers (Pasadena, 1963) (Proceedings of Symposia in Pure Mathematics), Volume 1965, American Mathematical Society, 1965, pp. 1-15 | Zbl

[37] Allan J. Silberger Introduction to harmonic analysis on reductive $p$ -adic groups, Mathematical Notes, 23, Princeton University Press, 1979, iv+372 pages

[38] Marko Tadic Harmonic analysis of spherical functions on reductive groups over $𝔭$ -adic fields, Pac. J. Math., Volume 109 (1983) no. 1, pp. 215-235 | DOI

[39] Jacques Tits Reductive groups over local fields, Automorphic forms, representations and $L$ -functions (Oregon State Univ., Corvallis, 1977) (Proceedings of Symposia in Pure Mathematics), Volume 33.1, American Mathematical Society, 1979, pp. 29-69 | DOI | Zbl

[40] Alessandro Veca The Kunze–Stein phenomenon, Ph. D. Thesis, Univ. of New South Wales (2002)

[41] Valentin E. Voskresensky Algebraic groups and their birational invariants, Translations of Mathematical Monographs, 179, American Mathematical Society, 1998, xiii+218 pages

[42] André Weil Adeles and algebraic groups, Progress in Mathematics, 23, Birkhäuser, 1982, v+126 pages | DOI

Cité par Sources :