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Arithmetic progressions -- an operator theoretic view

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  • 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.
    Mathematics Subject Classification: 47A35, 37A45.


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  • [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.


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


    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.


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


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


    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.


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


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


    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.


    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.


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


    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.


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


    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.


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


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


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


    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.


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


    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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