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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
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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 $i\le k$.

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Additional Information

V. Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
MR Author ID: 35155

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