Uniformity seminorms on ℓ∞ and applications
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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 ℤ/Nℤ 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 k ≥ 1, we define seminorms on ℓ∞(ℤ) 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.
- J. Auslander, Minimal Flows and their Extensions, North-Holland, Amsterdam, 1988.
- L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Princeton University Press, Princeton, 1963.
- J. Bourgain, H. Furstenberg, Y. Katznelson and D. Ornstein, Appendix on return-time sequences, Inst. Hautes Ètudes Sci. Publ. Math. 69 (1989), 42–45. CrossRef
- 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.
- V. Bergelson, B. Host and B. Kra, with an Appendix by I. Ruzsa, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), 261–303.
- A. Leibman and V. Bergelson, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), 155–230. CrossRef
- R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin Inc., New York, 1969.
- H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. CrossRef
- W. T. Gowers, A new proof of Szemerédi’s Theorem, Geom. Funct. Anal. 11 (2001), 465–588. CrossRef
- B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math (2) 167 (2008), 481–547.
- B. Green and T. Tao, Linear equations in the primes Ann. of Math, to appear, Available at: http://arxiv.org/abs/math/0606088
- B. Green and T. Tao, Quadratic uniformity of the Möbius function, Ann. Inst. Fourier 58 (2008), 1863–1935.
- B. Green and T. Tao, An inverse theorem for the Gowers U 3 -norm, with applications, Proc. Edinburgh Math. Soc. 51 (2008), 73–153. CrossRef
- B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.
- B. Host and B. Kra, Analysis of two step nilsequences, Ann. Inst. Fourier 58 (2008), 1407–1453.
- L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.
- A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213. CrossRef
- E. Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems 11 (1991), 379–391. CrossRef
- E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems 13 (1993), 767–784.
- M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer-Verlag, New York, 1987.
- N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415–426. CrossRef
- Uniformity seminorms on ℓ∞ and applications
Journal d'Analyse Mathématique
Volume 108, Issue 1 , pp 219-276
- Cover Date
- Print ISSN
- Online ISSN
- The Hebrew University Magnes Press
- Additional Links
- Author Affiliations
- 1. Laboratoire D’Analyse et de Mathématiques Appliquées, Université Paris-Est, UMR CNRS 8050, 5 BD Descartes, 77454, Marne La Vallée Cedex 2, France
- 2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA