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The exact convergence rate in the ergodic theorem of Lubotzky-Phillips-Sarnak and a universal lower bound on discrepancies. (English) Zbl 1492.37003

Summary: We establish a new result about the equidistribution of points on the two-dimensional round sphere. More precisely, we improve an upper bound of Lubotzky-Phillips-Sarnak on the discrepancies of some finite symmetric sets of isometries of the sphere defined with the help of Lipschitz quaternions. We show that a simple application of the spectral theorem leads to the best possible upper bound. Our proof relies on the deep result of Lubotzky-Phillips-Sarnak about spectral properties of some special free groups of isometries of the sphere. It leads to an upper bound on the discrepancy which matches exactly a general lower bound suspected by A. Lubotzky [Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Reprint of the 1994 original. Basel: Birkhäuser (2010; Zbl 1183.22001)]. We confirm Lubotzky’s guess by proving a universal lower bound on the discrepancies of actions on atomless probability spaces. We also mention some facts relating discrepancies to standard deviations, spectral gaps, amenability, Kazdhan pairs and \(\epsilon\)-good sets. In an appendix, we emphasize the relation between the computed discrepancies and the values of the Harish-Chandra function of a free group.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
22E40 Discrete subgroups of Lie groups
22D40 Ergodic theory on groups

Citations:

Zbl 1183.22001
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[1] V. I. A ’ and A. L. K , Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain. Dokl. Akad. Nauk SSSR 148 (1963), 9-12. Zbl 0237.34008 MR 0150374
[2] B. B , P. H and A. V , Kazhdan’s property (T). New Math. Monogr.
[3] V. I. B , Measure theory. Vol. I, II. Springer, Berlin, 2007. Zbl 1120.28001 MR 2267655 · Zbl 1120.28001
[4] M. B , Structure conforme au bord et flot géodésique d’un CAT. 1/-espace.
[5] Enseign. Math. (2) 41 (1995), no. 1-2, 63-102. Zbl 0871.58069 MR 1341941
[6] J. B and A. G , A spectral gap theorem in SU.d /. J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1455-1511. Zbl 1254.43010 MR 2966656 · Zbl 1254.43010
[7] J. B , Z. R and P. S , Spatial statistics for lattice points on the sphere I: Individual results. Bull. Iranian Math. Soc. 43 (2017), no. 4, 361-386. Zbl 7006293 MR 3711836 · Zbl 1464.11076
[8] J. B , P. S and Z. R , Local statistics of lattice points on the sphere.
[9] In Modern trends in constructive function theory, pp. 269-282, Contemp. Math. 661, Amer. Math. Soc., Providence, RI, 2016. Zbl 1394.11059 MR 3489563
[10] J. H. B , G. G , G. H and D. Z , The 1-2-3 of modular forms. Universitext, Springer, Berlin, 2008. Zbl 1197.11047 MR 2385372
[11] L. C , Automorphic forms and the distribution of points on odd-dimensional spheres. · Zbl 1087.11033
[12] Israel J. Math. 132 (2002), 175-187. Zbl 1087.11033 MR 1952619
[13] L. C , H. O and E. U , Hecke operators and equidistribution of Hecke points. Invent. Math. 144 (2001), no. 2, 327-351. Zbl 1144.11301 MR 1827734 · Zbl 1144.11301
[14] J. M. C , Operator norms on free groups. Boll. Un. Mat. Ital. B (6) 1 (1982), no. 3, 1055-1065. Zbl 0518.46050 MR 683492 · Zbl 0518.46050
[15] Y. C V , Distribution de points sur une sphère (d’après Lubotzky, Phillips et Sarnak). Exp. No. 703, 83-93, 177-178, 1989. Zbl 0701.11024 MR 1040569 · Zbl 0701.11024
[16] G. D , P. S and A. V , Elementary number theory, group theory, and Ramanujan graphs. London Math. Soc. Stud. Texts 55, Cambridge University Press, Cambridge, 2003. Zbl 1032.11001 MR 1989434 · Zbl 1032.11001
[17] P. H , On simplicity of reduced C -algebras of groups. Bull. Lond. Math. Soc. 39 (2007), no. 1, 1-26. Zbl 1123.22004 MR 2303514 · Zbl 1123.22004
[18] P. D , La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273-307. Zbl 0287.14001 MR 340258
[19] A. D and R. G , On spectra of Koopman, groupoid and quasi-regular representations. J. Mod. Dyn. 11 (2017), 99-123. Zbl 06991096 MR 3627119 · Zbl 1502.22003
[20] J. S. E , P. M and A. V , Linnik’s ergodic method and the distribution of integer points on spheres. In Automorphic representations and L-functions, pp. 119-185, Tata Inst. Fundam. Res. Stud. Math. 22, Tata Inst. Fund. Res., Mumbai, 2013. Zbl 1371.11071 MR 3156852 · Zbl 1371.11071
[21] V. F , Diophantine properties of groups of toral automorphisms, 2016, arXiv:1607.06019.
[22] U. F , The Ramanujan property for simplicial complexes, 2016, arXiv:1605.02664.
[23] O. G and Z. G , Explicit constructions of linear-sized superconcentrators. J. · Zbl 0487.05045
[24] Comput. System Sci. 22 (1981), no. 3, 407-420. Zbl 0487.05045 MR 633542 · Zbl 0487.05045
[25] R. G and V. S. V , Harmonic analysis of spherical functions on real reductive groups. Ergeb. Math. Grenzgeb. 101, Springer, Berlin, 1988. Zbl 0675.43004 MR 954385 · Zbl 0675.43004
[26] A. G and A. N , Quantitative ergodic theorems and their number-theoretic applications. Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 65-113. Zbl 1351.37039 MR 3286482 · Zbl 1351.37039
[27] U. H and S. K , A Lévy-Khinchin formula for free groups. Proc. Amer. Math. Soc. 143 (2015), no. 4, 1477-1489. Zbl 1311.43009 MR 3314063 · Zbl 1311.43009
[28] H -C , Two theorems on semi-simple Lie groups. Ann. of Math. (2) 83 (1966), 74-128. Zbl 0199.46403 MR 194556 · Zbl 0199.46403
[29] E. H and K. A. R , Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations. Grundlehren Math. Wiss. 115, Academic Press, New York; · Zbl 0837.43002
[30] Springer, Berlin, Göttingen, Heidelberg, 1963. Zbl 0837.43002 MR 0156915 · Zbl 0119.45901
[31] H. K , Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146-156. Zbl 0092.26704 MR 112053 · Zbl 0092.26704
[32] -Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336-354. Zbl 0092.33503 MR 109367 · Zbl 0092.33503
[33] M. G. K , Amenable actions and weak containment of certain representations of discrete groups. Proc. Amer. Math. Soc. 122 (1994), no. 3, 751-757. Zbl 0829.43003 MR 1209424 · Zbl 0829.43003
[34] A. L , Discrete groups, expanding graphs and invariant measures. Modern Birkhäuser Classics, Birkhäuser, Basel, 2010. Zbl 1183.22001 MR 2569682 · Zbl 1183.22001
[35] A. L , R. P and P. S , Hecke operators and distributing points on the sphere. I. pp. S149-S186, 39, 1986. Zbl 0619.10052 MR 861487 · Zbl 0619.10052
[36] -Hecke operators and distributing points on S 2 . II. Comm. Pure Appl. Math. 40 (1987), no. 4, 401-420. Zbl 0648.10034 MR 890171 · Zbl 0648.10034
[37] -Ramanujan graphs. Combinatorica 8 (1988), no. 3, 261-277. Zbl 0661.05035 MR 963118 · Zbl 0661.05035
[38] G. A. M , Discrete subgroups of semisimple Lie groups. Ergeb. Math. Grenzgeb.
[39] A. N , Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups. Math. Res. Lett. 5 (1998), no. 3, 305-325. Zbl 0942.22007 MR 1637840 · Zbl 0942.22007
[40] O. P and P. S , Super-golden-gates for P U.2/. Adv. Math. 327 (2018), 869-901. Zbl 1383.81059 MR 3762004 · Zbl 1383.81059
[41] A. P L , On the optimality of the Monte-Carlo estimator, 2019, arXiv:1903.06006.
[42] G. P , Quadratic forms in unitary operators. Linear Algebra Appl. 267 (1997), 125-137. Zbl 0889.47007 MR 1479116 · Zbl 0889.47007
[43] L. S -C and W. W , Transition operators on co-compact G-spaces. Rev.
[44] Mat. Iberoam. 22 (2006), no. 3, 747-799. Zbl 1116.22007 MR 2320401 · Zbl 1116.22007
[45] P. S , Some applications of modular forms. Cambridge Tracts in Math. 99, Cambridge University Press, Cambridge, 1990. Zbl 0721.11015 MR 1102679 · Zbl 0721.11015
[46] B. S , Mesure invariante et équirépartition dans les groupes compacts. In Autour du centenaire Lebesgue, pp. 63-84, Panor. Synthèses 18, Soc. Math. France, Paris, 2004. Zbl 1077.22003 MR 2143415 · Zbl 1077.22003
[47] Y. S , Random ergodic theorems, invariant means and unitary representation. In Lie groups and ergodic theory (Mumbai, 1996), pp. 273-314, Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res., Bombay, 1998. Zbl 0946.22007 MR 1699368 · Zbl 0946.22007
[48] -Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Ann. of Math. (2) 152 (2000), no. 1, 113-182. Zbl 0970.22011 MR 1792293 · Zbl 0970.22011
[49] A. W , L’intégration dans les groupes topologiques et ses applications. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann et Cie., Paris, 1940. Zbl 0063.08195 MR 0005741
[50] W. W , Random walks on infinite graphs and groups. Cambridge Tracts in Math. 138, Cambridge University Press, Cambridge, 2000. Zbl 0951.60002 MR 1743100 (Reçu le 3 novembre 2019) · Zbl 0951.60002
[51] Antoine P L , Aix Marseille Université, CNRS, Technopôle Château-Gombert, 39, rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France;
[52] Christophe P , Aix Marseille Université, CNRS, Technopôle Château-Gombert, 39, rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France, and Section de mathématiques, Université de Genève, rue du Conseil-Général 7-9, 1205
[53] Geneve, Switzerland; e-mail: pittet@math.cnrs
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