Nilpotent Structures in Ergodic Theory

Host, Bernard; Kra, Bryna

doi:10.1090/surv/236

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Nilpotent Structures in Ergodic Theory

About this Title

Bernard Host, Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France and Bryna Kra, Northwestern University, Evanston, IL

Publication: Mathematical Surveys and Monographs
Publication Year: 2018; Volume 236
ISBNs: 978-1-4704-4780-9 (print); 978-1-4704-5061-8 (online)
DOI: https://doi.org/10.1090/surv/236
MathSciNet review: MR3839640
MSC: Primary 37-02; Secondary 11B30, 28D05, 37Axx, 37B05, 47A35

PDF View full volume as PDF

Read more about this volume

O. Antolin Camarena and B. Szegedy, Nilspaces, nilmanifolds and their morphisms, arXiv: 1009.3825
Idris Assani, Wiener Wintner ergodic theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1995517, DOI 10.1142/4538
I. Assani, Averages along cubes for not necessarily commuting m.p.t, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 1–19. MR 2331322, DOI 10.1090/conm/430/08248
I. Assani, Pointwise convergence of ergodic averages along cubes, J. Anal. Math. 110 (2010), 241–269. MR 2753294, DOI 10.1007/s11854-010-0006-3
Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
L. Auslander, L. Green, and F. Hahn, Flows on homogeneous spaces, Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. MR 0167569
Tim Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 321–338. MR 2599882, DOI 10.1017/S014338570900011X
Tim Austin, Norm convergence of continuous-time polynomial multiple ergodic averages, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 361–382. MR 2901352, DOI 10.1017/S0143385711000563
Tim Austin, Pleasant extensions retaining algebraic structure, I, J. Anal. Math. 125 (2015), 1–36. MR 3317896, DOI 10.1007/s11854-015-0001-9
Tim Austin, Pleasant extensions retaining algebraic structure, II, J. Anal. Math. 126 (2015), 1–111. MR 3358029, DOI 10.1007/s11854-015-0013-5
Tim Austin, A proof of Walsh’s convergence theorem using couplings, Int. Math. Res. Not. IMRN 15 (2015), 6661–6674. MR 3384494, DOI 10.1093/imrn/rnu145
Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877, DOI 10.1017/CBO9780511735264
V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 337–349. MR 912373, DOI 10.1017/S0143385700004090
Vitaly Bergelson, Ergodic Ramsey theory—an update, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61. MR 1411215, DOI 10.1017/CBO9780511662812.002
Vitaly Bergelson, The multifarious Poincaré recurrence theorem, Descriptive set theory and dynamical systems (Marseille-Luminy, 1996) London Math. Soc. Lecture Note Ser., vol. 277, Cambridge Univ. Press, Cambridge, 2000, pp. 31–57. MR 1774423
Vitaly Bergelson, Hillel Furstenberg, and Benjamin Weiss, Piecewise-Bohr sets of integers and combinatorial number theory, Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, Berlin, 2006, pp. 13–37. MR 2249261, DOI 10.1007/3-540-33700-8_{2}
Vitaly Bergelson, Bernard Host, Randall McCutcheon, and François Parreau, Aspects of uniformity in recurrence. part 2, Colloq. Math. 84/85 (2000), no. part 2, 549–576. Dedicated to the memory of Anzelm Iwanik. MR 1784213, DOI 10.4064/cm-84/85-2-549-576
Vitaly Bergelson, Bernard Host, and Bryna Kra, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), no. 2, 261–303. With an appendix by Imre Ruzsa. MR 2138068, DOI 10.1007/s00222-004-0428-6
V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. 9 (1996), no. 3, 725–753. MR 1325795, DOI 10.1090/S0894-0347-96-00194-4
V. Bergelson and A. Leibman, Failure of the Roth theorem for solvable groups of exponential growth, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 45–53. MR 2041260, DOI 10.1017/S0143385703000427
V. Bergelson, A. Leibman, and E. Lesigne, Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory, J. Anal. Math. 103 (2007), 47–92. MR 2373264, DOI 10.1007/s11854-008-0002-z
Vitaly Bergelson and Alexander Leibman, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), no. 2, 155–230. MR 2318563, DOI 10.1007/s11511-007-0015-y
V. Bergelson, A. Leibman, and E. Lesigne, Intersective polynomials and the polynomial Szemerédi theorem, Adv. Math. 219 (2008), no. 1, 369–388. MR 2435427, DOI 10.1016/j.aim.2008.05.008
V. Bergelson, A. Leibman, and C. G. Moreira, From discrete- to continuous-time ergodic theorems, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 383–426. MR 2901353, DOI 10.1017/S0143385711000848
Vitaly Bergelson, Alexander Leibman, and Tamar Ziegler, The shifted primes and the multidimensional Szemerédi and polynomial van der Waerden theorems, C. R. Math. Acad. Sci. Paris 349 (2011), no. 3-4, 123–125 (English, with English and French summaries). MR 2769892, DOI 10.1016/j.crma.2010.11.028
V. Bergelson and R. McCutcheon, An ergodic IP polynomial Szemerédi theorem. Mem. Amer. Math. Soc. 146 (2000), no. 695
Vitaly Bergelson, Terence Tao, and Tamar Ziegler, An inverse theorem for the uniformity seminorms associated with the action of $\Bbb F^\infty _p$, Geom. Funct. Anal. 19 (2010), no. 6, 1539–1596. MR 2594614, DOI 10.1007/s00039-010-0051-1
Vitaly Bergelson, Terence Tao, and Tamar Ziegler, Multiple recurrence and convergence results associated to $\Bbb {F}_P^\omega$-actions, J. Anal. Math. 127 (2015), 329–378. MR 3421997, DOI 10.1007/s11854-015-0033-1
Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
G. Birkhoff, Proof of a recurrence theorem for strongly transitive systems, Proc. Nat. Acad. Sci. USA 17 (1931), 650–655.
G. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. USA 17 (1931), 656–660.
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
N. Bourbaki, Éléments de mathématique. Part I. Les structures fondamentales de l’analyse. Livre I. Théorie des ensembles. Actualités Scientifiques et Industrielles, no. 846. Hermann, Paris, 1939.
J. Bourgain, An approach to pointwise ergodic theorems, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 204–223. MR 950982, DOI 10.1007/BFb0081742
J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math. 404 (1990), 140–161. MR 1037434, DOI 10.1515/crll.1990.404.140
J. Bourgain, Pointwise ergodic theorems for arithmetic sets. With an appendix by J. Bourgain, H. Furstenberg, Y. Katznelson, and D. Ornstein. Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.
Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Pablo Candela, Notes on nilspaces: algebraic aspects, Discrete Anal. , posted on (2017), Paper No. 15, 59. MR 3695478, DOI 10.19086/da.2105
Pablo Candela, Notes on compact nilspaces, Discrete Anal. , posted on (2017), Paper No. 16, 57. MR 3695479, DOI 10.19086/da.2106
Pablo Candela and Olof Sisask, Convergence results for systems of linear forms on cyclic groups and periodic nilsequences, SIAM J. Discrete Math. 28 (2014), no. 2, 786–810. MR 3211031, DOI 10.1137/130935677
Qing Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1363–1369. MR 2465660, DOI 10.1090/S0002-9939-08-09614-7
Qing Chu, Convergence of multiple ergodic averages along cubes for several commuting transformations, Studia Math. 196 (2010), no. 1, 13–22. MR 2564479, DOI 10.4064/sm196-1-2
Qing Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems 31 (2011), no. 3, 771–792. MR 2794947, DOI 10.1017/S0143385710000258
Qing Chu and Nikos Frantzikinakis, Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 877–897. MR 2995648, DOI 10.1017/S014338571100006X
Qing Chu and Nikos Frantzikinakis, Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 877–897. MR 2995648, DOI 10.1017/S014338571100006X
Qing Chu, Nikos Frantzikinakis, and Bernard Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc. (3) 102 (2011), no. 5, 801–842. MR 2795725, DOI 10.1112/plms/pdq037
Qing Chu and Pavel Zorin-Kranich, Lower bound in the Roth theorem for amenable groups, Ergodic Theory Dynam. Systems 35 (2015), no. 6, 1746–1766. MR 3377282, DOI 10.1017/etds.2014.13
Jean-Pierre Conze and Emmanuel Lesigne, Théorèmes ergodiques pour des mesures diagonales, Bull. Soc. Math. France 112 (1984), no. 2, 143–175 (French, with English summary). MR 788966
Jean-Pierre Conze and Emmanuel Lesigne, Sur un théorème ergodique pour des mesures diagonales, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 12, 491–493 (French, with English summary). MR 939438
Jean-Pierre Conze and Emmanuel Lesigne, Sur un théorème ergodique pour des mesures diagonales, Probabilités, Publ. Inst. Rech. Math. Rennes, vol. 1987, Univ. Rennes I, Rennes, 1988, pp. 1–31 (French). MR 989141
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
Sebastián Donoso and Wenbo Sun, Dynamical cubes and a criteria for systems having product extensions, J. Mod. Dyn. 9 (2015), 365–405. MR 3446942, DOI 10.3934/jmd.2015.9.365
Sebastián Donoso and Wenbo Sun, A pointwise cubic average for two commuting transformations, Israel J. Math. 216 (2016), no. 2, 657–678. MR 3557461, DOI 10.1007/s11856-016-1423-5
Sebastián Donoso and Wenbo Sun, Pointwise multiple averages for systems with two commuting transformations, Ergodic Theory Dynam. Systems 38 (2018), no. 6, 2132–2157. MR 3833344, DOI 10.1017/etds.2016.127
Sebastián Donoso and Wenbo Sun, Pointwise convergence of some multiple ergodic averages, Adv. Math. 330 (2018), 946–996. MR 3787561, DOI 10.1016/j.aim.2018.03.022
Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
Tanja Eisner and Pavel Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete Contin. Dyn. Syst. 33 (2013), no. 8, 3497–3516. MR 3021367, DOI 10.3934/dcds.2013.33.3497
Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Robert Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271. MR 123635, DOI 10.1090/S0002-9947-1960-0123635-1
Robert Ellis and Harvey Keynes, A characterization of the equicontinuous structure relation, Trans. Amer. Math. Soc. 161 (1971), 171–183. MR 282357, DOI 10.1090/S0002-9947-1971-0282357-4
Nikos Frantzikinakis, The structure of strongly stationary systems, J. Anal. Math. 93 (2004), 359–388. MR 2110334, DOI 10.1007/BF02789313
Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems 26 (2006), no. 4, 1061–1071. MR 2246591, DOI 10.1017/S0143385706000204
Nikos Frantzikinakis, Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5435–5475. MR 2415080, DOI 10.1090/S0002-9947-08-04591-1
Nikos Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math. 109 (2009), 353–395. MR 2585398, DOI 10.1007/s11854-009-0035-y
Nikos Frantzikinakis, Multiple recurrence and convergence for Hardy sequences of polynomial growth, J. Anal. Math. 112 (2010), 79–135. MR 2762998, DOI 10.1007/s11854-010-0026-z
Nikos Frantzikinakis, A multidimensional Szemerédi theorem for Hardy sequences of different growth, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5653–5692. MR 3347186, DOI 10.1090/S0002-9947-2014-06275-2
Nikos Frantzikinakis, Multiple correlation sequences and nilsequences, Invent. Math. 202 (2015), no. 2, 875–892. MR 3418246, DOI 10.1007/s00222-015-0579-7
Nikos Frantzikinakis and Bernard Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc. 30 (2017), no. 1, 67–157. MR 3556289, DOI 10.1090/jams/857
Nikos Frantzikinakis and Bernard Host, Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 81–142. MR 3742539, DOI 10.1017/etds.2016.19
Nikos Frantzikinakis, Bernard Host, and Bryna Kra, Multiple recurrence and convergence for sequences related to the prime numbers, J. Reine Angew. Math. 611 (2007), 131–144. MR 2361086, DOI 10.1515/CRELLE.2007.076
Nikos Frantzikinakis, Bernard Host, and Bryna Kra, The polynomial multidimensional Szemerédi theorem along shifted primes, Israel J. Math. 194 (2013), no. 1, 331–348. MR 3047073, DOI 10.1007/s11856-012-0132-y
N. Frantzikinakis, M. Johnson, E. Lesigne, and M. Wierdl, Powers of sequences and convergence of ergodic averages, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1431–1456. MR 2718901, DOI 10.1017/S0143385709000571
Nikos Frantzikinakis and Bryna Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems 25 (2005), no. 3, 799–809. MR 2142946, DOI 10.1017/S0143385704000616
Nikos Frantzikinakis and Bryna Kra, Polynomial averages converge to the product of integrals, Israel J. Math. 148 (2005), 267–276. Probability in mathematics. MR 2191231, DOI 10.1007/BF02775439
Nikos Frantzikinakis and Bryna Kra, Ergodic averages for independent polynomials and applications, J. London Math. Soc. (2) 74 (2006), no. 1, 131–142. MR 2254556, DOI 10.1112/S0024610706023374
Nikos Frantzikinakis, Emmanuel Lesigne, and Máté Wierdl, Sets of $k$-recurrence but not $(k+1)$-recurrence, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 839–849 (English, with English and French summaries). MR 2266880
Nikos Frantzikinakis, Emmanuel Lesigne, and Máté Wierdl, Powers of sequences and recurrence, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 504–530. MR 2481957, DOI 10.1112/plms/pdn040
N. Frantzikinakis, E. Lesigne, and M. Wierdl, Random sequences and pointwise convergence of multiple ergodic averages, Indiana Univ. Math. J. 61 (2012), no. 2, 585–617. MR 3043589, DOI 10.1512/iumj.2012.61.4571
Nikos Frantzikinakis, Emmanuel Lesigne, and Máté Wierdl, Random differences in Szemerédi’s theorem and related results, J. Anal. Math. 130 (2016), 91–133. MR 3574649, DOI 10.1007/s11854-016-0030-z
Nikos Frantzikinakis and Máté Wierdl, A Hardy field extension of Szemerédi’s theorem, Adv. Math. 222 (2009), no. 1, 1–43. MR 2531366, DOI 10.1016/j.aim.2009.03.017
N. A. Friedman and D. S. Ornstein, On mixing and partial mixing, Illinois J. Math. 16 (1972), 61–68. MR 293059
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 157368, DOI 10.2307/2373137
Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI 10.1007/BF02813304
Harry Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 3, 211–234. MR 628658, DOI 10.1090/S0273-0979-1981-14932-6
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291 (1979). MR 531279, DOI 10.1007/BF02790016
H. Furstenberg, Y. Katznelson, and D. Ornstein, The ergodic theoretical proof of Szemerédi’s theorem, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 527–552. MR 670131, DOI 10.1090/S0273-0979-1982-15052-2
Hillel Furstenberg and Benjamin Weiss, A mean ergodic theorem for $(1/N)\sum ^N_{n=1}f(T^nx)g(T^{n^2}x)$, Convergence in ergodic theory and probability (Columbus, OH, 1993) Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 193–227. MR 1412607
Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
E. Glasner, $RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof. arXiv:1402.3135.
Morikuni Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc. 20 (1969), 157–162. MR 233923, DOI 10.1090/S0002-9939-1969-0233923-X
W. H. Gottschalk, Characterizations of almost periodic transformation groups, Proc. Amer. Math. Soc. 7 (1956), 709–712. MR 79754, DOI 10.1090/S0002-9939-1956-0079754-6
Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810
W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588. MR 1844079, DOI 10.1007/s00039-001-0332-9
W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. (2) 166 (2007), no. 3, 897–946. MR 2373376, DOI 10.4007/annals.2007.166.897
W. T. Gowers, Decompositions, approximate structure, transference, and the Hahn-Banach theorem, Bull. Lond. Math. Soc. 42 (2010), no. 4, 573–606. MR 2669681, DOI 10.1112/blms/bdq018
Ben Green and Terence Tao, An inverse theorem for the Gowers $U^3(G)$ norm, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 1, 73–153. MR 2391635, DOI 10.1017/S0013091505000325
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379, DOI 10.4007/annals.2008.167.481
Benjamin Green and Terence Tao, Linear equations in primes, Ann. of Math. (2) 171 (2010), no. 3, 1753–1850. MR 2680398, DOI 10.4007/annals.2010.171.1753
Ben Green and Terence Tao, An arithmetic regularity lemma, an associated counting lemma, and applications, An irregular mind, Bolyai Soc. Math. Stud., vol. 21, János Bolyai Math. Soc., Budapest, 2010, pp. 261–334. MR 2815606, DOI 10.1007/978-3-642-14444-8_{7}
Ben Green and Terence Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), no. 2, 541–566. MR 2877066, DOI 10.4007/annals.2012.175.2.3
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), no. 2, 465–540. MR 2877065, DOI 10.4007/annals.2012.175.2.2
Ben Green and Terence Tao, On the quantitative distribution of polynomial nilsequences—erratum [MR2877065], Ann. of Math. (2) 179 (2014), no. 3, 1175–1183. MR 3171762, DOI 10.4007/annals.2014.179.3.8
Ben Green, Terence Tao, and Tamar Ziegler, An inverse theorem for the Gowers $U^4$-norm, Glasg. Math. J. 53 (2011), no. 1, 1–50. MR 2747135, DOI 10.1017/S0017089510000546
Ben Green, Terence Tao, and Tamar Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Ann. of Math. (2) 176 (2012), no. 2, 1231–1372. MR 2950773, DOI 10.4007/annals.2012.176.2.11
Y. Gutman, F. Manners, and P. Varjú, The structure theory of Nilspaces I., J. Anal. Math., to appear.
Y. Gutman, F. Manners, and P. Varjú, The structure theory of Nilspaces II: Representation as nilmanifolds, Trans. Amer. Math. Soc., to appear.
Y. Gutman, F. Manners, & P. Varjú, The structure theory of Nilspaces III: Inverse limit representations and topological dynamics, arXiv:1605.08950
P. Hall, A Contribution to the Theory of Groups of Prime-Power Order, Proc. London Math. Soc. (2) 36 (1934), 29–95. MR 1575964, DOI 10.1112/plms/s2-36.1.29
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
F. J. Hahn, On affine transformations of compact abelian groups, Amer. J. Math. 85 (1963), 428–446. MR 155956, DOI 10.2307/2373133
Bernard Host, Progressions arithmétiques dans les nombres premiers (d’après B. Green et T. Tao), Astérisque 307 (2006), Exp. No. 944, viii, 229–246 (French, with French summary). Séminaire Bourbaki. Vol. 2004/2005. MR 2296420
Bernard Host, Ergodic seminorms for commuting transformations and applications, Studia Math. 195 (2009), no. 1, 31–49. MR 2539560, DOI 10.4064/sm195-1-3
Bernard Host and Bryna Kra, Convergence of Conze-Lesigne averages, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 493–509. MR 1827115, DOI 10.1017/S0143385701001249
Bernard Host and Bryna Kra, An odd Furstenberg-Szemerédi theorem and quasi-affine systems, J. Anal. Math. 86 (2002), 183–220. MR 1894481, DOI 10.1007/BF02786648
Bernard Host and Bryna Kra, Averaging along cubes, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 123–144. MR 2090768
Bernard Host and Bryna Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), no. 1, 397–488. MR 2150389, DOI 10.4007/annals.2005.161.397
Bernard Host and Bryna Kra, Convergence of polynomial ergodic averages, Israel J. Math. 149 (2005), 1–19. Probability in mathematics. MR 2191208, DOI 10.1007/BF02772534
Bernard Host and Bryna Kra, Parallelepipeds, nilpotent groups and Gowers norms, Bull. Soc. Math. France 136 (2008), no. 3, 405–437 (English, with English and French summaries). MR 2415348, DOI 10.24033/bsmf.2561
Bernard Host and Bryna Kra, Analysis of two step nilsequences, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 5, 1407–1453 (English, with English and French summaries). MR 2445824
Bernard Host and Bryna Kra, Uniformity seminorms on $\ell ^\infty$ and applications, J. Anal. Math. 108 (2009), 219–276. MR 2544760, DOI 10.1007/s11854-009-0024-1
Bernard Host and Bryna Kra, Nil-Bohr sets of integers, Ergodic Theory Dynam. Systems 31 (2011), no. 1, 113–142. MR 2755924, DOI 10.1017/S014338570900087X
Bernard Host and Bryna Kra, A point of view on Gowers uniformity norms, New York J. Math. 18 (2012), 213–248. MR 2920990
Bernard Host, Bryna Kra, and Alejandro Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math. 224 (2010), no. 1, 103–129. MR 2600993, DOI 10.1016/j.aim.2009.11.009
Bernard Host, Bryna Kra, and Alejandro Maass, Complexity of nilsystems and systems lacking nilfactors, J. Anal. Math. 124 (2014), 261–295. MR 3286054, DOI 10.1007/s11854-014-0032-7
Bernard Host and Alejandro Maass, Nilsystèmes d’ordre 2 et parallélépipèdes, Bull. Soc. Math. France 135 (2007), no. 3, 367–405 (French, with English and French summaries). MR 2430186, DOI 10.24033/bsmf.2539
Wen Huang, Song Shao, and Xiangdong Ye, Nil Bohr-sets and almost automorphy of higher order, Mem. Amer. Math. Soc. 241 (2016), no. 1143, v+83. MR 3476203, DOI 10.1090/memo/1143
W. Huang, S. Shao, and X. Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, J. d’Analyse Math., to appear.
W. Huang, S. Shao, and X. Ye, Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages, arXiv:1312.7213.
Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548
Michael C. R. Johnson, Convergence of polynomial ergodic averages of several variables for some commuting transformations, Illinois J. Math. 53 (2009), no. 3, 865–882 (2010). MR 2727359
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. U. S. A. 18 (1932), 255–263.
Andreas Koutsogiannis, Integer part polynomial correlation sequences, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1525–1542. MR 3789175, DOI 10.1017/etds.2016.67
Andreas Koutsogiannis, Closest integer polynomial multiple recurrence along shifted primes, Ergodic Theory Dynam. Systems 38 (2018), no. 2, 666–685. MR 3774837, DOI 10.1017/etds.2016.40
Bryna Kra, The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 1, 3–23. MR 2188173, DOI 10.1090/S0273-0979-05-01086-4
Bryna Kra, From combinatorics to ergodic theory and back again, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 57–76. MR 2275670
Bryna Kra, Ergodic methods in additive combinatorics, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 103–143. MR 2359470, DOI 10.1090/crmp/043/07
Bryna Kra, Poincaré recurrence and number theory: thirty years later [comment on the reprint of MR0628658], Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 4, 497–501. MR 2823016, DOI 10.1090/S0273-0979-2011-01343-X
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496
A. Le. Nilsequences and multiple correlations along subsequences. Ergodic Theory Dynam. Systems, to appear.
A. Leibman, Polynomial sequences in groups, J. Algebra 201 (1998), no. 1, 189–206. MR 1608723, DOI 10.1006/jabr.1997.7269
A. Leibman, Polynomial mappings of groups, Israel J. Math. 129 (2002), 29–60. MR 1910931, DOI 10.1007/BF02773152
A. Leibman, Lower bounds for ergodic averages, Ergodic Theory Dynam. Systems 22 (2002), no. 3, 863–872. MR 1908558, DOI 10.1017/S0143385702000433
A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 201–213. MR 2122919, DOI 10.1017/S0143385704000215
A. Leibman, Pointwise convergence of ergodic averages for polynomial actions of ${\Bbb Z}^d$ by translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 215–225. MR 2122920, DOI 10.1017/S0143385704000227
A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math. 146 (2005), 303–315. MR 2151605, DOI 10.1007/BF02773538
A. Leibman, Rational sub-nilmanifolds of a compact nilmanifold, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 787–798. MR 2237471, DOI 10.1017/S014338570500057X
A. Leibman, Orbits on a nilmanifold under the action of a polynomial sequence of translations, Ergodic Theory Dynam. Systems 27 (2007), no. 4, 1239–1252. MR 2342974, DOI 10.1017/S0143385706000940
A. Leibman, Orbit of the diagonal in the power of a nilmanifold, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1619–1658. MR 2563743, DOI 10.1090/S0002-9947-09-04961-7
A. Leibman, Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 841–854. MR 2643713, DOI 10.1017/S0143385709000303
A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems 35 (2015), no. 1, 176–191. MR 3294297, DOI 10.1017/etds.2013.36
Emmanuel Lesigne, Résolution d’une équation fonctionnelle, Bull. Soc. Math. France 112 (1984), no. 2, 177–196 (French, with English summary). MR 788967
Emmanuel Lesigne, Théorèmes ergodiques pour une translation sur un nilvariété, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 115–126 (French, with English summary). MR 991492, DOI 10.1017/S0143385700004843
E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems 10 (1990), no. 3, 513–521 (French, with English summary). MR 1074316, DOI 10.1017/S014338570000571X
Emmanuel Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 379–391 (French, with English summary). MR 1116647, DOI 10.1017/S0143385700006209
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk SSSR. Ser. Mat. 13 (1949), 9–32 (Russian). MR 0028842
Douglas C. McMahon, Relativized weak disjointness and relatively invariant measures, Trans. Amer. Math. Soc. 236 (1978), 225–237. MR 467704, DOI 10.1090/S0002-9947-1978-0467704-9
J. Moreira and F. Richter, A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems, Ergodic Theory Dynam. Systems, to appear.
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Robert T. Moore, Measurable, continuous and smooth vectors for semi-groups and group representations, Memoirs of the American Mathematical Society, No. 78, American Mathematical Society, Providence, R.I., 1968. MR 0229091
Calvin C. Moore and Klaus Schmidt, Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson, Proc. London Math. Soc. (3) 40 (1980), no. 3, 443–475. MR 572015, DOI 10.1112/plms/s3-40.3.443
Brendan Nagle, Vojtěch Rödl, and Mathias Schacht, The counting lemma for regular $k$-uniform hypergraphs, Random Structures Algorithms 28 (2006), no. 2, 113–179. MR 2198495, DOI 10.1002/rsa.20117
William Parry, Compact abelian group extensions of discrete dynamical systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 95–113. MR 260976, DOI 10.1007/BF00537014
William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350
William Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc. 2 (1970), 37–40. MR 267558, DOI 10.1112/blms/2.1.37
William Parry, Dynamical representations in nilmanifolds, Compositio Math. 26 (1973), 159–174. MR 320277
K. R. Parthasarathy, Introduction to probability and measure, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0651012
Julian Petresco, Sur les commutateurs, Math. Z. 61 (1954), 348–356 (French). MR 66380, DOI 10.1007/BF01181351
Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728
B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of Math. (2) 52 (1950), 293–308. MR 38358, DOI 10.2307/1969471
Amanda Potts, Multiple ergodic averages for flows and an application, Illinois J. Math. 55 (2011), no. 2, 589–621 (2012). MR 3020698
Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI 10.1215/S0012-7094-91-06311-8
Walter Rudin, Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1038803, DOI 10.1002/9781118165621
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Daniel J. Rudolph, Eigenfunctions of $T\times S$ and the Conze-Lesigne algebra, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993) London Math. Soc. Lecture Note Ser., vol. 205, Cambridge Univ. Press, Cambridge, 1995, pp. 369–432. MR 1325712, DOI 10.1017/CBO9780511574818.017
Daniel J. Rudolph, Fully generic sequences and a multiple-term return-times theorem, Invent. Math. 131 (1998), no. 1, 199–228. MR 1489899, DOI 10.1007/s002220050202
A. Sárközy, On difference sets of sequences of integers. III, Acta Math. Acad. Sci. Hungar. 31 (1978), no. 3-4, 355–386. MR 487031, DOI 10.1007/BF01901984
Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
Nimish A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J. 75 (1994), no. 3, 711–732. MR 1291701, DOI 10.1215/S0012-7094-94-07521-2
Song Shao and Xiangdong Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math. 231 (2012), no. 3-4, 1786–1817. MR 2964624, DOI 10.1016/j.aim.2012.07.012
S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545, DOI 10.1007/978-3-642-85473-6
Wenbo Sun, Multiple recurrence and convergence for certain averages along shifted primes, Ergodic Theory Dynam. Systems 35 (2015), no. 5, 1592–1609. MR 3365735, DOI 10.1017/etds.2014.6
B. Szegedy. On higher order Fourier analysis. arXiv: 1203.2260
E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245. MR 369312, DOI 10.4064/aa-27-1-199-245
Terence Tao, A quantitative ergodic theory proof of Szemerédi’s theorem, Electron. J. Combin. 13 (2006), no. 1, Research Paper 99, 49. MR 2274314, DOI 10.37236/1125
Terence Tao, The dichotomy between structure and randomness, arithmetic progressions, and the primes, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 581–608. MR 2334204, DOI 10.4171/022-1/22
Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 657–688. MR 2408398, DOI 10.1017/S0143385708000011
Terence Tao, Structure and randomness, American Mathematical Society, Providence, RI, 2008. Pages from year one of a mathematical blog. MR 2459552, DOI 10.1090/mbk/059
Terence Tao, Poincaré’s legacies, pages from year two of a mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. MR 2523047, DOI 10.1090/mbk/066
T. Tao, Deducing a weak ergodic inverse theorem from a combinatorial inverse theorem. What’s new blog, https://terrytao.wordpress.com/2015/07/23.
Terence Tao and Tamar Ziegler, The primes contain arbitrarily long polynomial progressions, Acta Math. 201 (2008), no. 2, 213–305. MR 2461509, DOI 10.1007/s11511-008-0032-5
Terence Tao and Tamar Ziegler, The inverse conjecture for the Gowers norm over finite fields in low characteristic, Ann. Comb. 16 (2012), no. 1, 121–188. MR 2948765, DOI 10.1007/s00026-011-0124-3
Terence Tao and Tamar Ziegler, A multi-dimensional Szemerédi theorem for the primes via a correspondence principle, Israel J. Math. 207 (2015), no. 1, 203–228. MR 3358045, DOI 10.1007/s11856-015-1157-9
Henry Towsner, Convergence of diagonal ergodic averages, Ergodic Theory Dynam. Systems 29 (2009), no. 4, 1309–1326. MR 2529651, DOI 10.1017/S0143385708000722
Siming Tu and Xiangdong Ye, Dynamical parallelepipeds in minimal systems, J. Dynam. Differential Equations 25 (2013), no. 3, 765–776. MR 3095274, DOI 10.1007/s10884-013-9313-6
J. G. van der Corput, Neue zahlentheoretische Abschätzungen, Math. Z. 29 (1929), no. 1, 397–426 (German). MR 1545013, DOI 10.1007/BF01180539
J. G. van der Corput, Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins, Acta Math. 56 (1931), no. 1, 373–456 (German). MR 1555330, DOI 10.1007/BF02545780
William A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. Math. 90 (1968), 723–732. MR 232377, DOI 10.2307/2373480
J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), 263–266.
Miguel N. Walsh, Norm convergence of nilpotent ergodic averages, Ann. of Math. (2) 175 (2012), no. 3, 1667–1688. MR 2912715, DOI 10.4007/annals.2012.175.3.15
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Benjamin Weiss, Single orbit dynamics, CBMS Regional Conference Series in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2000. MR 1727510, DOI 10.1090/cbms/095
H. Weyl, Über ein Problem aus dem Gebiete der diophantischen Approximationen, Nachr. Ges. Wiss. Gottingen. Math.-phys. KI, 1914, 234–244.
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415–426. MR 4098, DOI 10.2307/2371534
Máté Wierdl, Pointwise ergodic theorem along the prime numbers, Israel J. Math. 64 (1988), no. 3, 315–336 (1989). MR 995574, DOI 10.1007/BF02882425
Trevor D. Wooley and Tamar D. Ziegler, Multiple recurrence and convergence along the primes, Amer. J. Math. 134 (2012), no. 6, 1705–1732. MR 2999293, DOI 10.1353/ajm.2012.0048
Hidehiko Yamabe, On an arcwise connected subgroup of a Lie group, Osaka Math. J. 2 (1950), 13–14. MR 36766
T. Ziegler, A non-conventional ergodic theorem for a nilsystem, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1357–1370. MR 2158410, DOI 10.1017/S0143385703000518
Tamar Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2007), no. 1, 53–97. MR 2257397, DOI 10.1090/S0894-0347-06-00532-7
Robert J. Zimmer, Extensions of ergodic actions and generalized discrete spectrum, Bull. Amer. Math. Soc. 81 (1975), 633–636. MR 372160, DOI 10.1090/S0002-9904-1975-13770-0
Robert J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20 (1976), no. 4, 555–588. MR 414832
Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373–409. MR 409770
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4
Pavel Zorin-Kranich, Cube spaces and the multiple term return times theorem, Ergodic Theory Dynam. Systems 34 (2014), no. 5, 1747–1760. MR 3255440, DOI 10.1017/etds.2013.9

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Nilpotent Structures in Ergodic Theory

About this Title

Table of Contents

References [Enhancements On Off] (What's this?)