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Quantitative ergodic theorems and their number-theoretic applications

Authors: Alexander Gorodnik and Amos Nevo
Journal: Bull. Amer. Math. Soc. 52 (2015), 65-113
MSC (2010): Primary 37A15, 37P55, 22E46, 11J83, 11F70
Published electronically: June 11, 2014
MathSciNet review: 3286482
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Abstract: We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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  • [Aa] Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400 (99d:28025)
  • [AAB] Claire Anantharaman, Jean-Philippe Anker, Martine Babillot, Aline Bonami, Bruno Demange, Sandrine Grellier, François Havard, Philippe Jaming, Emmanuel Lesigne, Patrick Maheux, Jean-Pierre Otal, Barbara Schapira, and Jean-Pierre Schreiber, Théorèmes ergodiques pour les actions de groupes, Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], vol. 41, L'Enseignement Mathématique, Geneva, 2010 (French). With a foreword in English by Amos Nevo. MR 2643350 (2011h:47010)
  • [Ar] Vladimir I. Arnold, Arnold's problems, Springer-Verlag, Berlin; PHASIS, Moscow, 2004. Translated and revised edition of the 2000 Russian original; With a preface by V. Philippov, A. Yakivchik and M. Peters. MR 2078115 (2005c:58001)
  • [AK63] V. I. Arnold and A. L. Krylov, 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 (Russian). MR 0150374 (27 #375)
  • [B02] Martine Babillot, Points entiers et groupes discrets: de l'analyse aux systèmes dynamiques, Rigidité, groupe fondamental et dynamique, Panor. Synthèses, vol. 13, Soc. Math. France, Paris, 2002, pp. 1-119 (French, with English and French summaries). With an appendix by Emmanuel Breuillard. MR 1993148 (2004i:37057)
  • [B82] Hans-Jochen Bartels, Nichteuklidische Gitterpunktprobleme und Gleichverteilung in linearen algebraischen Gruppen, Comment. Math. Helv. 57 (1982), no. 1, 158-172 (German). MR 672852 (84c:22013),
  • [BHV] B. Bekka, P. de la Harpe and A. Valette, Kazhdan's property (T). New Mathematical Monographs, 11. Cambridge University Press, Cambridge, 2008.
  • [BO12] Yves Benoist and Hee Oh, Effective equidistribution of $ S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 5, 1889-1942 (English, with English and French summaries). MR 3025156,
  • [B31] G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. USA 17 (1931), 656-660.
  • [BB13] Valentin Blomer and Farrell Brumley, The role of the Ramanujan conjecture in analytic number theory, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 2, 267-320. MR 3020828,
  • [BW80] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917 (83c:22018)
  • [BF11] Jean Bourgain and Elena Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc. 24 (2011), no. 4, 945-967. MR 2813334 (2012d:11072),
  • [Bo83] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69-85. MR 874045 (88f:42036),
  • [Bo89] Jean Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5-45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein. MR 1019960 (90k:28030)
  • [BFLM11] Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, J. Amer. Math. Soc. 24 (2011), no. 1, 231-280. MR 2726604 (2011k:37008),
  • [BGS06] Jean Bourgain, Alex Gamburd, and Peter Sarnak, Sieving and expanders, C. R. Math. Acad. Sci. Paris 343 (2006), no. 3, 155-159 (English, with English and French summaries). MR 2246331 (2007b:11139),
  • [BGS10] Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559-644. MR 2587341 (2011d:11018),
  • [BK10] Jean Bourgain and Alex Kontorovich, On representations of integers in thin subgroups of $ {\rm SL}_2(\mathbb{Z})$, Geom. Funct. Anal. 20 (2010), no. 5, 1144-1174. MR 2746949 (2012i:11008),
  • [BMW99] R. W. Bruggeman, R. J. Miatello, and N. R. Wallach, Resolvent and lattice points on symmetric spaces of strictly negative curvature, Math. Ann. 315 (1999), no. 4, 617-639. MR 1731464 (2001f:11170),
  • [BGM11] Roelof Wichert Bruggeman, Fritz Grunewald, and Roberto Jorge Miatello, New lattice point asymptotics for products of upper half-planes, Int. Math. Res. Not. IMRN 7 (2011), 1510-1559. MR 2806513 (2012e:11095),
  • [BS91] M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), no. 1, 1-11. MR 1123369 (92m:22005),
  • [C53] A. P. Calderon, A general ergodic theorem, Ann. of Math. (2) 58 (1953), 182-191. MR 0055415 (14,1071a)
  • [CT10] Antoine Chambert-Loir and Yuri Tschinkel, Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math. 2 (2010), no. 3, 351-429. MR 2740045 (2012d:11143),
  • [C03] Laurent Clozel, Démonstration de la conjecture $ \tau $, Invent. Math. 151 (2003), no. 2, 297-328 (French). MR 1953260 (2004f:11049),
  • [CW76] Ronald R. Coifman and Guido Weiss, Transference methods in analysis, American Mathematical Society, Providence, R.I., 1976. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31. MR 0481928 (58 #2019)
  • [C] B. Conrad, Modular forms and the Ramanujan conjecture, Cambridge University Press, 2011.
  • [C78] Michael Cowling, The Kunze-Stein phenomenon, Ann. Math. (2) 107 (1978), no. 2, 209-234. MR 0507240 (58 #22398)
  • [C79] Michael Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976-1978), II, Lecture Notes in Math., vol. 739, Springer, Berlin, 1979, pp. 132-178 (French). MR 560837 (81e:22019)
  • [CN01] M. Cowling and A. Nevo, Uniform estimates for spherical functions on complex semisimple Lie groups, Geom. Funct. Anal. 11 (2001), no. 5, 900-932. MR 1873133 (2002k:43005),
  • [D85] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55-89; erratum: J. Reine Angew. Math. 359 (1985), 214.
  • [D86] S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986), no. 4, 636-660. MR 870710 (88i:22011),
  • [D89] S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, Number theory and dynamical systems (York, 1987) London Math. Soc. Lecture Note Ser., vol. 134, Cambridge Univ. Press, Cambridge, 1989, pp. 69-86. MR 1043706 (91d:58200),
  • [D71] Pierre Deligne, Formes modulaires et représentations $ l$-adiques, Séminaire Bourbaki. Vol. 1968/69: Exposés 347-363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 355, 139-172 (French). MR 3077124
  • [D42] Jean Delsarte, Sur le gitter fuchsien, C. R. Acad. Sci. Paris 214 (1942), 147-179 (French). MR 0007769 (4,191a)
  • [DH] Harold G. Diamond and H. Halberstam, A higher-dimensional sieve method, Cambridge Tracts in Mathematics, vol. 177, Cambridge University Press, Cambridge, 2008. With an appendix (``Procedures for computing sieve functions'') by William F. Galway. MR 2458547 (2009h:11151)
  • [Du03] W. Duke, Rational points on the sphere, Ramanujan J. 7 (2003), no. 1-3, 235-239. Rankin memorial issues. MR 2035804 (2005h:11094),
  • [DRS93] W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143-179. MR 1230289 (94k:11072),
  • [EG74] Frederick P. Greenleaf and William R. Emerson, Group structure and the pointwise ergodic theorem for connected amenable groups, Advances in Math. 14 (1974), 153-172. MR 0384997 (52 #5867)
  • [EM93] Alex Eskin and Curt McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), no. 1, 181-209. MR 1230290 (95b:22025),
  • [EMS96] Alex Eskin, Shahar Mozes, and Nimish Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2) 143 (1996), no. 2, 253-299. MR 1381987 (97d:22012),
  • [F06] John B. Friedlander, Producing prime numbers via sieve methods, Analytic number theory, Lecture Notes in Math., vol. 1891, Springer, Berlin, 2006, pp. 1-49. MR 2277657 (2007k:11153),
  • [FI] John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984 (2011d:11227)
  • [GJ78] Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $ {\rm GL}(2)$ and $ {\rm GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471-542. MR 533066 (81e:10025)
  • [GGN1] Anish Ghosh, Alexander Gorodnik, and Amos Nevo, Diophantine approximation and automorphic spectrum, Int. Math. Res. Not. IMRN 21 (2013), 5002-5058. MR 3123673
  • [GGN2] A. Ghosh, A. Gorodnik and A. Nevo, Metric Diophantine approximation on homogeneous varieties. Math arXiv:1205.4426. To appear in Compositio Mathematica.
  • [GGN3] A. Ghosh, A. Gorodnik and A. Nevo, Best possible rate of distribution of dense lattice orbits on homogeneous varieties. In preparation.
  • [GGN4] A. Ghosh, A. Gorodnik and A. Nevo, Diophantine approximation exponents on homogeneous varieties. To appear in Contemporary Math, 2014.
  • [G83] Anton Good, Local analysis of Selberg's trace formula, Lecture Notes in Mathematics, vol. 1040, Springer-Verlag, Berlin, 1983. MR 727476 (85k:11026)
  • [G03] Alexander Gorodnik, Lattice action on the boundary of $ {\rm SL}(n,\mathbb{R})$, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1817-1837. MR 2032490 (2005c:22015),
  • [G04] Alexander Gorodnik, Uniform distribution of orbits of lattices on spaces of frames, Duke Math. J. 122 (2004), no. 3, 549-589. MR 2057018 (2005f:37009),
  • [GM05] Alexander Gorodnik and Francois Maucourant, Proximality and equidistribution on the Furstenberg boundary, Geom. Dedicata 113 (2005), 197-213. MR 2171305 (2006e:37015),
  • [GN10] Alexander Gorodnik and Amos Nevo, The ergodic theory of lattice subgroups, Annals of Mathematics Studies, vol. 172, Princeton University Press, Princeton, NJ, 2010. MR 2573139 (2011c:22006)
  • [GN12a] Alexander Gorodnik and Amos Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127-176. MR 2889708,
  • [GN12b] Alexander Gorodnik and Amos Nevo, On Arnold's and Kazhdan's equidistribution problems, Ergodic Theory Dynam. Systems 32 (2012), no. 6, 1972-1990. MR 2995880,
  • [GN12c] A. Gorodnik and A. Nevo, Lifting, restricting and sifting integral points on affine homogeneous varieties, Compositio Math. 2012, doi:10.1112/S0010437X12000516.
  • [GN] Alexander Gorodnik and Amos Nevo, Ergodic theory and the duality principle on homogeneous spaces, Geom. Funct. Anal. 24 (2014), no. 1, 159-244. MR 3177381,
  • [GW07] Alex Gorodnik and Barak Weiss, Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal. 17 (2007), no. 1, 58-115. MR 2306653 (2008i:37012),
  • [G] George Greaves, Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43, Springer-Verlag, Berlin, 2001. MR 1836967 (2002i:11092)
  • [Gr10] Andrew Granville, Different approaches to the distribution of primes, Milan J. Math. 78 (2010), no. 1, 65-84. MR 2684773 (2011g:11178),
  • [Gu10] Antonin Guilloux, Polynomial dynamic and lattice orbits in $ S$-arithmetic homogeneous spaces, Confluentes Math. 2 (2010), no. 1, 1-35. MR 2649235 (2011i:37003),
  • [G69] Yves Guivarch, Généralisation d'un théorème de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1020-A1023 (French). MR 0251191 (40 #4422)
  • [HR] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730 (54 #12689)
  • [H] Glyn Harman, Prime-detecting sieves, London Mathematical Society Monographs Series, vol. 33, Princeton University Press, Princeton, NJ, 2007. MR 2331072 (2008k:11097)
  • [HP01] Sa'ar Hersonsky and Frédéric Paulin, Hausdorff dimension of Diophantine geodesics in negatively curved manifolds, J. Reine Angew. Math. 539 (2001), 29-43. MR 1863852 (2002i:53053),
  • [HP02a] Sa'ar Hersonsky and Frédéric Paulin, Diophantine approximation in negatively curved manifolds and in the Heisenberg group, Rigidity in dynamics and geometry (Cambridge, 2000) Springer, Berlin, 2002, pp. 203-226. MR 1919402 (2003i:11112)
  • [HP02b] Sa'ar Hersonsky and Frédéric Paulin, Diophantine approximation for negatively curved manifolds, Math. Z. 241 (2002), no. 1, 181-226. MR 1930990 (2003g:53051),
  • [HM79] Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), no. 1, 72-96. MR 533220 (80g:22017),
  • [HT] Roger Howe and Eng-Chye Tan, Nonabelian harmonic analysis, Universitext, Springer-Verlag, New York, 1992. Applications of $ {\rm SL}(2,{\bf R})$. MR 1151617 (93f:22009)
  • [H56] Heinz Huber, Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I, Comment. Math. Helv. 30 (1956), 20-62 (1955) (German). MR 0074536 (17,603b)
  • [I78] Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Invent. Math. 47 (1978), no. 2, 171-188. MR 0485740 (58 #5553)
  • [I96] Henryk Iwaniec, The lowest eigenvalue for congruence groups, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 203-212. MR 1390315 (97e:11058)
  • [IK] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • [JL70] H. Jacquet and R. P. Langlands, Automorphic forms on $ {\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654 (53 #5481)
  • [Jo93] Roger L. Jones, Ergodic averages on spheres, J. Anal. Math. 61 (1993), 29-45. MR 1253437 (95g:28031),
  • [JR79] Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), no. 3, 185-197. MR 553340 (81d:10042),
  • [KW82] Yitzhak Katznelson and Benjamin Weiss, A simple proof of some ergodic theorems, Israel J. Math. 42 (1982), no. 4, 291-296. MR 682312 (84i:28020),
  • [K65] D. A. Každan, Uniform distribution on a plane, Trudy Moskov. Mat. Obšč. 14 (1965), 299-305 (Russian). MR 0193187 (33 #1408)
  • [K67] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71-74 (Russian). MR 0209390 (35 #288)
  • [KS03] H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, appendix in H. Kim, Functoriality for the exterior square of GL(4) and the symmetric fourth of GL(2). J. Amer. Math. Soc. 16 (2003), no. 1, 139-183.
  • [K98a] Dmitry Y. Kleinbock, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J. 95 (1998), no. 1, 107-124. MR 1646538 (99k:11107),
  • [K98b] Dmitry Y. Kleinbock, Bounded orbits conjecture and Diophantine approximation, Lie groups and ergodic theory (Mumbai, 1996) Tata Inst. Fund. Res. Stud. Math., vol. 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 119-130. MR 1699361 (2000k:11089)
  • [K99] Dmitry Kleinbock, Badly approximable systems of affine forms, J. Number Theory 79 (1999), no. 1, 83-102. MR 1724255 (2001b:11064),
  • [K01] Dmitry Kleinbock, Some applications of homogeneous dynamics to number theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 639-660. MR 1858548 (2002g:37009),
  • [K10] Dmitry Kleinbock, Quantitative nondivergence and its Diophantine applications, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 131-153. MR 2648694 (2011m:11144)
  • [KM98] D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148 (1998), no. 1, 339-360. MR 1652916 (99j:11083),
  • [KM99] D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), no. 3, 451-494. MR 1719827 (2001i:37046),
  • [KM] D. Kleinbock and K. Merrill, Rational approximation on spheres. arXiv:1301.0989.
  • [K09] Alex V. Kontorovich, The hyperbolic lattice point count in infinite volume with applications to sieves, Duke Math. J. 149 (2009), no. 1, 1-36. MR 2541126 (2011f:11125),
  • [K] Alex Kontorovich, From Apollonius to Zaremba: local-global phenomena in thin orbits, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 2, 187-228. MR 3020826,
  • [KO11] Alex Kontorovich and Hee Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc. 24 (2011), no. 3, 603-648. With an appendix by Oh and Nimish Shah. MR 2784325,
  • [K1] E. Kowalski, The large sieve and its applications, Cambridge Tracts in Mathematics, vol. 175, Cambridge University Press, Cambridge, 2008. Arithmetic geometry, random walks and discrete groups. MR 2426239 (2009f:11123)
  • [K2] E. Kowalski: Sieve in expansion, Séminaire Bourbaki, Exposé 1028 (November 2010)
  • [K3] E. Kowalski, Sieves in discrete groups, especially sparse. Math. arXiv, 1207.7051, January 2013.
  • [KO12] Alex Kontorovich and Hee Oh, Almost prime Pythagorean triples in thin orbits, J. Reine Angew. Math. 667 (2012), 89-131. MR 2929673
  • [KS60] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $ 2\times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1-62. MR 0163988 (29 #1287)
  • [K69] Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. MR 0245725 (39 #7031)
  • [La95] Michael T. Lacey, Ergodic averages on circles, J. Anal. Math. 67 (1995), 199-206. MR 1383493 (97f:28045),
  • [L89] Steven P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), no. 1-2, 1-55. MR 1007619 (91c:58112),
  • [L65] Serge Lang, Report on Diophantine approximations, Bull. Soc. Math. France 93 (1965), 177-192. MR 0193064 (33 #1286)
  • [LP82] Peter D. Lax and Ralph S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), no. 3, 280-350. MR 661875 (83j:10057),
  • [L01] Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), no. 2, 259-295. MR 1865397 (2002h:37005),
  • [L99] François Ledrappier, Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 61-64 (French, with English and French summaries). MR 1703338 (2000c:22009),
  • [LP03] F. Ledrappier and M. Pollicott, Ergodic properties of linear actions of $ (2\times 2)$-matrices, Duke Math. J. 116 (2003), no. 2, 353-388. MR 1953296 (2003j:37041),
  • [L95] Jian-Shu Li, The minimal decay of matrix coefficients for classical groups, Harmonic analysis in China, Math. Appl., vol. 327, Kluwer Acad. Publ., Dordrecht, 1995, pp. 146-169. MR 1355801 (98d:22009)
  • [LZ96] Jian-Shu Li and Chen-Bo Zhu, On the decay of matrix coefficients for exceptional groups, Math. Ann. 305 (1996), no. 2, 249-270. MR 1391214 (97f:22029),
  • [L44a] U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139-178 (English, with Russian summary). MR 0012111 (6,260b)
  • [L44b] U. V. Linnik, On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 347-368 (English, with Russian summary). MR 0012112 (6,260c)
  • [LS10] Jianya Liu and Peter Sarnak, Integral points on quadrics in three variables whose coordinates have few prime factors, Israel J. Math. 178 (2010), 393-426. MR 2733075 (2011k:11052),
  • [Lu12] Alexander Lubotzky, Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 1, 113-162. MR 2869010 (2012m:05003),
  • [LPS86] A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere. I, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149-S186. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861487 (88m:11025a),
  • [LPS87] A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on $ S^2$. II, Comm. Pure Appl. Math. 40 (1987), no. 4, 401-420. MR 890171 (88m:11025b),
  • [LRS95] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387-401. MR 1334872 (96h:11045),
  • [Mag02] Akos Magyar, Diophantine equations and ergodic theorems, Amer. J. Math. 124 (2002), no. 5, 921-953. MR 1925339 (2003f:37015)
  • [M02] Gregory Margulis, Diophantine approximation, lattices and flows on homogeneous spaces, A panorama of number theory or the view from Baker's garden (Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, pp. 280-310. MR 1975458 (2004h:11031),
  • [M04] Grigoriy A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska. MR 2035655 (2004m:37049)
  • [MNS00] G. A. Margulis, A. Nevo, and E. M. Stein, Analogs of Wiener's ergodic theorems for semisimple Lie groups. II, Duke Math. J. 103 (2000), no. 2, 233-259. MR 1760627 (2001h:22008),
  • [Ma02] F. Maucourant, Approximation diophantienne, dynamique des chambres de Weyl et répartitions d'orbites de réseaux, PhD Thesis, Université de Lille, 2002.
  • [Ma07] François Maucourant, Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices, Duke Math. J. 136 (2007), no. 2, 357-399. MR 2286635 (2007j:22022),
  • [MW11] François Maucourant and Barak Weiss, Lattice actions on the plane revisited, Geom. Dedicata 157 (2012), 1-21. MR 2893477,
  • [MW92] R. Miatello and N. R. Wallach, The resolvent of the Laplacian on locally symmetric spaces, J. Differential Geom. 36 (1992), no. 3, 663-698. MR 1189500 (93i:58160)
  • [Mol10] R. A. Mollin, An overview of sieve methods, Int. J. Contemp. Math. Sci. 5 (2010), no. 1-4, 67-80. MR 2667834
  • [Mo08] Yoichi Motohashi, An overview of sieve methods and their history [translation of Sūgaku 57 (2005), no. 2, 138-163; MR2142054], Sugaku Expositions 21 (2008), no. 1, 1-32. Sugaku Expositions. MR 2406271
  • [N94] Amos Nevo, Harmonic analysis and pointwise ergodic theorems for noncommuting transformations, J. Amer. Math. Soc. 7 (1994), no. 4, 875-902. MR 1266737 (95h:22006),
  • [N94b] Amos Nevo, Pointwise ergodic theorems for radial averages on simple Lie groups. I, Duke Math. J. 76 (1994), no. 1, 113-140. MR 1301188 (96c:28027),
  • [N97] Amos Nevo, Pointwise ergodic theorems for radial averages on simple Lie groups. II, Duke Math. J. 86 (1997), no. 2, 239-259. MR 1430433 (98m:28041),
  • [N98] Amos Nevo, Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett. 5 (1998), no. 3, 305-325. MR 1637840 (99e:28030),
  • [N06] Amos Nevo, Pointwise ergodic theorems for actions of groups, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 871-982. MR 2186253 (2006k:37006),
  • [NS10] Amos Nevo and Peter Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), no. 2, 361-402. MR 2746350 (2011m:22040),
  • [NS94] Amos Nevo and Elias M. Stein, A generalization of Birkhoff's pointwise ergodic theorem, Acta Math. 173 (1994), no. 1, 135-154. MR 1294672 (95m:28025),
  • [NS97] Amos Nevo and Elias M. Stein, Analogs of Wiener's ergodic theorems for semisimple groups. I, Ann. of Math. (2) 145 (1997), no. 3, 565-595. MR 1454704 (98m:22007),
  • [NT97] Amos Nevo and Sundaram Thangavelu, Pointwise ergodic theorems for radial averages on the Heisenberg group, Adv. Math. 127 (1997), no. 2, 307-334. MR 1448717 (98f:22005),
  • [N03] Markus Neuhauser, Kazhdan constants and matrix coefficients of $ {\rm Sp}(n,{\bf R})$, J. Lie Theory 13 (2003), no. 1, 133-154. MR 1958578 (2003m:22015)
  • [N02] Arnaldo Nogueira, Orbit distribution on $ \mathbb{R}^2$ under the natural action of $ {\rm SL}(2,\mathbb{Z})$, Indag. Math. (N.S.) 13 (2002), no. 1, 103-124. MR 2014978 (2005h:22034),
  • [N10] Arnaldo Nogueira, Lattice orbit distribution on $ \mathbb{R}^2$, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1201-1214. MR 2669417 (2011h:11076),
  • [O98] Hee Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France 126 (1998), no. 3, 355-380 (English, with English and French summaries). Erratum, MR 1682805 (2000b:22015)
  • [O02] Hee Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002), no. 1, 133-192. MR 1905394 (2003d:22015),
  • [O10] Hee Oh, Orbital counting via mixing and unipotent flows, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 339-375. MR 2648698 (2011f:11079)
  • [O11] H. Oh, Harmonic analysis, ergodic theory and counting for thin groups. arXiv:1208.4148.
  • [O14] Hee Oh, Apollonian circle packings: dynamics and number theory, Jpn. J. Math. 9 (2014), no. 1, 69-97. MR 3173439,
  • [P76] S. J. Patterson, A lattice-point problem in hyperbolic space, Mathematika 22 (1975), no. 1, 81-88. MR 0422160 (54 #10152)
  • [P94] Adam Parusiński, Subanalytic functions, Trans. Amer. Math. Soc. 344 (1994), no. 2, 583-595. MR 1160156 (94k:32006),
  • [P95] Mark Pollicott, A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds, Amer. J. Math. 117 (1995), no. 2, 289-305. MR 1323676 (96k:58169),
  • [R38] Frederick Riesz, Some mean ergodic theorems, J. London Math. Soc. S1-13, no. 4, 274. MR 1574977,
  • [R] 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, 125. Birkhaäuser Verlag, Basel, 1994.
  • [SV] A. Salehi Golsefidy and Péter P. Varjú, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832-1891. MR 3000503,
  • [SS] Alireza Salehi Golsefidy and Peter Sarnak, The affine sieve, J. Amer. Math. Soc. 26 (2013), no. 4, 1085-1105. MR 3073885,
  • [S95] Peter Sarnak, Selberg's eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272-1277. MR 1355461 (97c:11059)
  • [S03] Peter Sarnak, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441-478. MR 1997348 (2004f:11107),
  • [S05] Peter Sarnak, Notes on the generalized Ramanujan conjectures, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 659-685. MR 2192019 (2007a:11067)
  • [S07] P. Sarnak, Letter to Lagarias on integral Apollonian packings,
  • [S08] P. Sarnak, Equidistribution and primes,
  • [S09] P. Sarnak, Integral Apollonian Packings,
  • [S81] Klaus Schmidt, Amenability, Kazhdan's property $ (T)$, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynamical Systems 1 (1981), no. 2, 223-236. MR 661821 (83m:43001)
  • [Sch08] Eric Schmutz, Rational points on the unit sphere, Cent. Eur. J. Math. 6 (2008), no. 3, 482-487. MR 2425007 (2009c:11112),
  • [S56] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87. MR 0088511 (19,531g)
  • [S65] Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1-15. MR 0182610 (32 #93)
  • [St61] E. M. Stein, On the maximal ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1894-1897. MR 0131517 (24 #A1367)
  • [SW78] Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239-1295. MR 508453 (80k:42023),
  • [T67] A. A. Tempelman, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR 176 (1967), 790-793 (Russian). MR 0219700 (36 #2779)
  • [T92] Arkady Tempelman, Ergodic theorems for group actions, Mathematics and its Applications, vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Informational and thermodynamical aspects; Translated and revised from the 1986 Russian original. MR 1172319 (94f:22007)
  • [V] P. Varjú, Random walks in Euclidean space. arXiv:1205.3399.
  • [vN32] J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), 70-82.
  • [W99] Michel Waldschmidt, Density measure of rational points on abelian varieties, Nagoya Math. J. 155 (1999), 27-53. MR 1711387 (2000h:11077)
  • [W39] Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1-18. MR 1546100,

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Additional Information

Alexander Gorodnik
Affiliation: School of Mathematics, University of Bristol, Bristol, United Kingdom

Amos Nevo
Affiliation: Department of Mathematics, Technion, Israel

Received by editor(s): April 25, 2013
Published electronically: June 11, 2014
Additional Notes: The first author was supported in part by EPSRC, ERC, and RCUK
The second author was supported by an ISF grant
Article copyright: © Copyright 2014 American Mathematical Society

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