Geodesic
Combinatorics/Dynamical Systems
The shifted primes and the multidimensional Szemerédi and polynomial Van der Waerden theorems
[Translatés de l'ensemble des nombres premiers, théorème de Szemerédi multidimensionnel et théorème de Van der Waerden polynomial multidimensionnel]

Présenté par : Christophe Soulé

Bergelson, Vitaly1 ; Leibman, Alexander1 ; Ziegler, Tamar2
1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States
2 Department of Mathematics, Technion, Haifa 32000, Israel
Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 123-125.

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In this short note we establish new refinements of multidimensional Szemerédi and polynomial Van der Waerden theorems along the shifted primes.

Nous présentons de nouveaux résultats du type Szemerédi multidimensionnel et Van der Waerden polynomial multidimensionnel le long des ensembles P−1 et P+1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.11.028

Bergelson, Vitaly 1 ; Leibman, Alexander 1 ; Ziegler, Tamar 2

1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States
2 Department of Mathematics, Technion, Haifa 32000, Israel 
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Bergelson, Vitaly; Leibman, Alexander; Ziegler, Tamar. The shifted primes and the multidimensional Szemerédi and polynomial Van der Waerden theorems. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 123-125. doi : 10.1016/j.crma.2010.11.028. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.11.028/

[1] Bergelson, V. Ergodic theory and Diophantine problems, Temuco, 1997 (London Math. Soc. Lecture Note Ser.), Volume vol. 279, Cambridge Univ. Press, Cambridge (2000), pp. 167-205

[2] Bergelson, V.; Leibman, A. Polynomial Van der Waerden and Szemerédi theorems, J. Amer. Math. Soc., Volume 9 (1996), pp. 725-753

[3] Bergelson, V.; Leibman, A. Set-polynomials and polynomial extension of the Hales–Jewett theorem, Ann. of Math. (2), Volume 150 (1999) no. 1, pp. 33-75

[4] Bergelson, V.; McCutcheon, R. Recurrence for semigroup actions and a non-commutative Schur theorem, Minneapolis, MN, 1995 (Contemp. Math.), Volume vol. 215, Amer. Math. Soc., Providence, RI (1998), pp. 205-222

[5] Frantzikinakis, N.; Host, B.; Kra, B. Multiple recurrence and convergence for sequences related to the prime numbers, J. Reine Angew. Math., Volume 611 (2007), pp. 131-144

[6] Furstenberg, H.; Katznelson, Y. An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Anal. Math., Volume 45 (1985), pp. 117-168

[7] Furstenberg, H.; Weiss, B. Topological dynamics and combinatorial number theory, J. Anal. Math., Volume 34 (1978), pp. 61-85

[8] Green, B.; Tao, T. Linear equations in primes, Ann. of Math. (2), Volume 171 (2010) no. 3, pp. 1753-1850

[9] B. Green, T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., in press.

[10] B. Green, T. Tao, T. Ziegler, An inverse theorem for the Gowers Us+1[N] norm, preprint.

[11] McCutcheon, R. Elemental Methods in Ergodic Ramsey Theory, Lecture Notes in Math., vol. 1722, Springer-Verlag, Berlin, 1999

[12] T. Wooley, T. Ziegler, Multiple recurrence and convergence along the primes, Amer. J. Math., in press.

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☆ The first and the third authors are supported by BSF grant No. 2006094. The first and the second authors are supported by NSF grant DMS-0901106.