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]
Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 123-125.

Nous présentons de nouveaux résultats du type Szemerédi multidimensionnel et Van der Waerden polynomial multidimensionnel le long des ensembles P1 et P+1.

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

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://www.numdam.org/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.

Cité par Sources :

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.