Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Polynomial extensions of van der Waerden's
and Szemerédi's theorems

Authors: V. Bergelson and A. Leibman
Journal: J. Amer. Math. Soc. 9 (1996), 725-753
MSC (1991): Primary 11B83, 28D05, 54H20; Secondary 05A17, 05D10
MathSciNet review: 1325795
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let $S\subseteq \mathbb Z^l$ be a set of positive upper Banach density, let $p_1(n),\dotsc ,p_k(n)$ be polynomials with rational coefficients taking integer values on the integers and satisfying $p_i(0)=0$, $i=1,\dotsc ,k;$ then for any $ v_1,\dotsc ,v_k\in \mathbb Z^l$ there exist an integer $n$ and a vector $u\in \mathbb Z^l$ such that $u+p_i(n)v_i\in S$ for each $i\le k$.

References [Enhancements On Off] (What's this?)

  • [B1] V. Bergelson, Ergodic Ramsey theory, Cont. Math. 65 (1987), 63--87.MR 88h:05003
  • [B2] ------, Weakly mixing PET, Ergod. Th. and Dynam. Sys. 7 (1987), 337--349. MR 89g:28022
  • [BPT] A. Blaszczyk, S. Plewik and S. Turek, Topological multidimensional van der Waerden theorem, Comment. Math. Univ. Carolinae 30, N4 (1989), 783--787. MR 92a:54035
  • [F1] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d'Analyse Math. 31 (1977), 204--256. MR 58:16583
  • [F2] ------, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. MR 82j:28010
  • [FK1] H. Furstenberg, and Y. Katznelson, An ergodic Szemerédi thoerem for commuting transformations, J. d'Analyse Math. 34 (1978), 275--291. MR 82c:28032
  • [FK2] ------, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. d'Analyse Math. 45 (1985), 117--168. MR 87m:28007
  • [FK3] ------, A density version of the Hales-Jewett theorem, J. d'Analyse Math. 57 (1991), 64--119; Discrete Math. 75 (1989), 227--241. MR 90k:05003
  • [FKO] H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. 7 (1982), 527--552. MR 84b:28016
  • [FW] H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. d'Analyse Math. 34 (1978), 61--85. MR 80g:05009
  • [S] E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199--245. MR 51:5547

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 11B83, 28D05, 54H20, 05A17, 05D10

Retrieve articles in all journals with MSC (1991): 11B83, 28D05, 54H20, 05A17, 05D10

Additional Information

V. Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

A. Leibman
Affiliation: Department of Mathematics, Technion, Haifa 23000, Israel
Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): June 8, 1994
Received by editor(s) in revised form: March 30, 1995
Additional Notes: The first author gratefully acknowledges support received from the National Science Foundation (USA) via grants DMS-9103056 and DMS-9401093. The second author was supported by the British Technion Society.
Article copyright: © Copyright 1996 American Mathematical Society