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Uniformity seminorms on and applications. (English) Zbl 1183.37011
Summary: A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on /N introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k1, we define seminorms on () analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.
MSC:
37A30Ergodic theorems, spectral theory, Markov operators
37A05Measure-preserving transformations
37A45Relations of ergodic theory with number theory and harmonic analysis
20D15Nilpotent finite groups, p-groups
References:
[1]J. Auslander, Minimal Flows and their Extensions, North-Holland, Amsterdam, 1988.
[2]L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Princeton University Press, Princeton, 1963.
[3]J. Bourgain, H. Furstenberg, Y. Katznelson and D. Ornstein, Appendix on return-time sequences, Inst. Hautes Ètudes Sci. Publ. Math. 69 (1989), 42–45. · doi:10.1007/BF02698839
[4]V. Bergelson, H. Furstenberg and B. Weiss, Piecewise-Bohr sets of integers and combinatorial number theory, Algorithms Combin. 26, Springer, Berlin, 2006, pp. 13–37.
[5]V. Bergelson, B. Host and B. Kra, with an Appendix by I. Ruzsa, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), 261–303.
[6]A. Leibman and V. Bergelson, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), 155–230. · Zbl 1137.37005 · doi:10.1007/s11511-007-0015-y
[7]R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin Inc., New York, 1969.
[8]H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. · Zbl 0347.28016 · doi:10.1007/BF02813304
[9]W. T. Gowers, A new proof of Szemerédi’s Theorem, Geom. Funct. Anal. 11 (2001), 465–588. · Zbl 1028.11005 · doi:10.1007/s00039-001-0332-9
[10]B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math (2) 167 (2008), 481–547.
[11]B. Green and T. Tao, Linear equations in the primes Ann. of Math, to appear, Available at: http://arxiv.org/abs/math/0606088
[12]B. Green and T. Tao, Quadratic uniformity of the Möbius function, Ann. Inst. Fourier 58 (2008), 1863–1935.
[13]B. Green and T. Tao, An inverse theorem for the Gowers U 3 -norm, with applications, Proc. Edinburgh Math. Soc. 51 (2008), 73–153. · Zbl 1202.11013 · doi:10.1017/S0013091505000325
[14]B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.
[15]B. Host and B. Kra, Analysis of two step nilsequences, Ann. Inst. Fourier 58 (2008), 1407–1453.
[16]L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.
[17]A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213. · Zbl 1080.37003 · doi:10.1017/S0143385704000215
[18]E. Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems 11 (1991), 379–391. · Zbl 0709.28012 · doi:10.1017/S0143385700006209
[19]E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems 13 (1993), 767–784.
[20]M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer-Verlag, New York, 1987.
[21]N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415–426. · doi:10.2307/2371534