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June  2013, 6(3): 657-667. doi: 10.3934/dcdss.2013.6.657

Arithmetic progressions -- an operator theoretic view

Tanja Eisner 1, and  Rainer Nagel 2,

1. 

KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands
2. 

University of Tübingen, Mathematics Institute, Auf der Morgenstelle 10, D-72076 Tübingen

Received  February 2010 Revised  May 2010 Published  December 2012

Motivated by the recent Green--Tao theorem on arithmetic progressions in the primes, we discuss some of the basic operator theoretic techniques used in its proof. In particular, we obtain a complete proof of Szemerédi's theorem for arithmetic progressions of length $3$ (Roth's theorem) and the Furstenberg--Sárközy theorem.
Keywords: Roth's theorem, Jacobs-Glicksberg-deLeeuw decomposition., multiple ergodic theorems, Arithmetic progressions, operator theoretic methods.
Mathematics Subject Classification: 47A35, 37A4.
Citation: Tanja Eisner, Rainer Nagel. Arithmetic progressions -- an operator theoretic view. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 657-667. doi: 10.3934/dcdss.2013.6.657
References:
[1]

V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.  Google Scholar

[2]

V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and the polynomial Szemerédi theorem, Adv. Math., 219 (2008), 369-388.  Google Scholar

[3]

M. Einsiedler and T. Ward, "Ergodic Theory: With a View Towards Number Theory," Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

[4]

T. Eisner, "Stability of Operators and Operator Semigroups," Birkhäuser Verlag, Basel, 2010.  Google Scholar

[5]

T. Eisner, B. Farkas, M. Haase and R. Nagel, "Operator Theoretic Aspects of Ergodic Theory," Graduate Texts in Mathematics, Springer, 2013. Google Scholar

[6]

T. Eisner, B. Farkas, R. Nagel and A. Serény, Weakly and almost weakly stable $C_0$-semigroups, Int. J. Dyn. Syst. Differ. Equ., 1 (2007), 44-57. doi: 10.1504/IJDSDE.2007.013744.  Google Scholar

[7]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," Princeton University Press, Princeton, New Jersey, 1981.  Google Scholar

[8]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  Google Scholar

[9]

H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc., 7 (1982), 527-552. doi: 10.1090/S0273-0979-1982-15052-2.  Google Scholar

[10]

H. Furstenberg and B. Weiss, A mean ergodic theorem for $\frac{1}N sum_{n=1}^N f(T^nx) g(T^{n^2}x)$, Convergence in Ergodic Theory and Probability, eds: Bergelson, March, Rosenblatt, Walter de Gruyter & Co, Berlin, New York, (1996), 193-227.  Google Scholar

[11]

B. Green, "Lectures on Ergodic Theory, Part III,", , ().   Google Scholar

[12]

B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals Math., 167 (2008), 481-547. doi: 10.4007/annals.2008.167.481.  Google Scholar

[13]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals Math., 161 (2005), 397-488. doi: 10.4007/annals.2005.161.397.  Google Scholar

[14]

B. Kra, The Green-Tao Theorem on arithmetic progressions in the primes: An ergodic point of view, Bull. Amer. Math. Soc., 43 (2006), 3-23. doi: 10.1090/S0273-0979-05-01086-4.  Google Scholar

[15]

B. Kra, Ergodic methods in additive combinatorics, Additive combinatorics, 103-143, CRM Proc. Lecture Notes, 43, Amer. Math. Soc., Providence, RI, (2007).  Google Scholar

[16]

K. Petersen, "Ergodic Theory," Cambridge University Press, 1983.  Google Scholar

[17]

H. H. Schaefer, "Banach Lattices and Positive Operators," Springer-Verlag, 1974.  Google Scholar

[18]

T. Tao, The dichotomy between structure and randomness, arithmetic progressions, and the primes, International Congress of Mathematicians, I 581-608, Eur. Math. Soc., Zürich, (2007). doi: 10.4171/022-1/22.  Google Scholar

[19]

T. Tao, "Topics in Ergodic Theory,", 2008, ().   Google Scholar

[20]

T. Tao, "The Van der Corput Trick, and Equidistribution on Nilmanifolds," in Topics in Ergodic Theory, 2008, http://terrytao.wordpress.com/2008/06/14/the-van-der-corputs-trick-and-equidistribution-on-nilmanifolds. Google Scholar

show all references

References:
[1]

V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.  Google Scholar

[2]

V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and the polynomial Szemerédi theorem, Adv. Math., 219 (2008), 369-388.  Google Scholar

[3]

M. Einsiedler and T. Ward, "Ergodic Theory: With a View Towards Number Theory," Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

[4]

T. Eisner, "Stability of Operators and Operator Semigroups," Birkhäuser Verlag, Basel, 2010.  Google Scholar

[5]

T. Eisner, B. Farkas, M. Haase and R. Nagel, "Operator Theoretic Aspects of Ergodic Theory," Graduate Texts in Mathematics, Springer, 2013. Google Scholar

[6]

T. Eisner, B. Farkas, R. Nagel and A. Serény, Weakly and almost weakly stable $C_0$-semigroups, Int. J. Dyn. Syst. Differ. Equ., 1 (2007), 44-57. doi: 10.1504/IJDSDE.2007.013744.  Google Scholar

[7]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," Princeton University Press, Princeton, New Jersey, 1981.  Google Scholar

[8]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  Google Scholar

[9]

H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc., 7 (1982), 527-552. doi: 10.1090/S0273-0979-1982-15052-2.  Google Scholar

[10]

H. Furstenberg and B. Weiss, A mean ergodic theorem for $\frac{1}N sum_{n=1}^N f(T^nx) g(T^{n^2}x)$, Convergence in Ergodic Theory and Probability, eds: Bergelson, March, Rosenblatt, Walter de Gruyter & Co, Berlin, New York, (1996), 193-227.  Google Scholar

[11]

B. Green, "Lectures on Ergodic Theory, Part III,", , ().   Google Scholar

[12]

B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals Math., 167 (2008), 481-547. doi: 10.4007/annals.2008.167.481.  Google Scholar

[13]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals Math., 161 (2005), 397-488. doi: 10.4007/annals.2005.161.397.  Google Scholar

[14]

B. Kra, The Green-Tao Theorem on arithmetic progressions in the primes: An ergodic point of view, Bull. Amer. Math. Soc., 43 (2006), 3-23. doi: 10.1090/S0273-0979-05-01086-4.  Google Scholar

[15]

B. Kra, Ergodic methods in additive combinatorics, Additive combinatorics, 103-143, CRM Proc. Lecture Notes, 43, Amer. Math. Soc., Providence, RI, (2007).  Google Scholar

[16]

K. Petersen, "Ergodic Theory," Cambridge University Press, 1983.  Google Scholar

[17]

H. H. Schaefer, "Banach Lattices and Positive Operators," Springer-Verlag, 1974.  Google Scholar

[18]

T. Tao, The dichotomy between structure and randomness, arithmetic progressions, and the primes, International Congress of Mathematicians, I 581-608, Eur. Math. Soc., Zürich, (2007). doi: 10.4171/022-1/22.  Google Scholar

[19]

T. Tao, "Topics in Ergodic Theory,", 2008, ().   Google Scholar

[20]

T. Tao, "The Van der Corput Trick, and Equidistribution on Nilmanifolds," in Topics in Ergodic Theory, 2008, http://terrytao.wordpress.com/2008/06/14/the-van-der-corputs-trick-and-equidistribution-on-nilmanifolds. Google Scholar

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