# American Institute of Mathematical Sciences

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 & 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.   Google Scholar [2] V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and the polynomial Szemerédi theorem,, Adv. Math., 219 (2008), 369.   Google Scholar [3] M. Einsiedler and T. Ward, "Ergodic Theory: With a View Towards Number Theory,", Springer-Verlag London, (2011).  doi: 10.1007/978-0-85729-021-2.  Google Scholar [4] T. Eisner, "Stability of Operators and Operator Semigroups,", Birkhäuser Verlag, (2010).   Google Scholar [5] T. Eisner, B. Farkas, M. Haase and R. Nagel, "Operator Theoretic Aspects of Ergodic Theory,", Graduate Texts in Mathematics, (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.  doi: 10.1504/IJDSDE.2007.013744.  Google Scholar [7] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", Princeton University Press, (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.   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.  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, (1996), 193.   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.  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.  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.  doi: 10.1090/S0273-0979-05-01086-4.  Google Scholar [15] B. Kra, Ergodic methods in additive combinatorics,, Additive combinatorics, 43 (2007), 103.   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 (2007), 581.  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).   Google Scholar

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##### 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.   Google Scholar [2] V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and the polynomial Szemerédi theorem,, Adv. Math., 219 (2008), 369.   Google Scholar [3] M. Einsiedler and T. Ward, "Ergodic Theory: With a View Towards Number Theory,", Springer-Verlag London, (2011).  doi: 10.1007/978-0-85729-021-2.  Google Scholar [4] T. Eisner, "Stability of Operators and Operator Semigroups,", Birkhäuser Verlag, (2010).   Google Scholar [5] T. Eisner, B. Farkas, M. Haase and R. Nagel, "Operator Theoretic Aspects of Ergodic Theory,", Graduate Texts in Mathematics, (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.  doi: 10.1504/IJDSDE.2007.013744.  Google Scholar [7] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", Princeton University Press, (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.   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.  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, (1996), 193.   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.  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.  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.  doi: 10.1090/S0273-0979-05-01086-4.  Google Scholar [15] B. Kra, Ergodic methods in additive combinatorics,, Additive combinatorics, 43 (2007), 103.   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 (2007), 581.  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).   Google Scholar
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