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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 |
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,", , ().
|
[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] | |
[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,", , ().
|
[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] | |
[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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