## 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

## 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

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## Additional Information

**V. Bergelson**- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
- MR Author ID: 35155
- Email: vitaly@math.ohio-state.edu
**A. Leibman**- Affiliation: Department of Mathematics, Technion, Haifa 23000, Israel
- Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305
- Email: sashal@techunix.technion.ac.il, leibman@math.stanford.edu
- 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: https://doi.org/10.1090/S0894-0347-96-00194-4
- MathSciNet review: 1325795