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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
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by V. Bergelson and A. Leibman PDF
J. Amer. Math. Soc. 9 (1996), 725-753 Request permission


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 $i\le k$.
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Additional Information
  • V. Bergelson
  • Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
  • MR Author ID: 35155
  • Email:
  • A. Leibman
  • Affiliation: Department of Mathematics, Technion, Haifa 23000, Israel
  • Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305
  • Email:,
  • 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.
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 725-753
  • MSC (1991): Primary 11B83, 28D05, 54H20; Secondary 05A17, 05D10
  • DOI:
  • MathSciNet review: 1325795