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

Arithmetic progressions -- an operator theoretic view

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.
Citation: Tanja Eisner, Rainer Nagel. Arithmetic progressions -- an operator theoretic view. Discrete and 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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[11]

B. Green, "Lectures on Ergodic Theory, Part III," http://www.dpmms.cam.ac.uk/ bjg23/ergodic-theory.html.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[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.

[19]

T. Tao, "Topics in Ergodic Theory," 2008, http://terrytao.wordpress.com/category/254a-ergodic-theory/.

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[11]

B. Green, "Lectures on Ergodic Theory, Part III," http://www.dpmms.cam.ac.uk/ bjg23/ergodic-theory.html.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[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.

[19]

T. Tao, "Topics in Ergodic Theory," 2008, http://terrytao.wordpress.com/category/254a-ergodic-theory/.

[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.

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