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

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 $i\le k$.