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 ⊆ Z l S\subseteq \mathbb {Z}^l be a set of positive upper Banach density, let p 1 ( n ) , … , p k ( n ) p_1(n),\dotsc ,p_k(n) be polynomials with rational coefficients taking integer values on the integers and satisfying p i ( 0 ) = 0 p_i(0)=0 , i = 1 , … , k ; i=1,\dotsc ,k; then for any v 1 , … , v k ∈ Z l v_1,\dotsc ,v_k\in \mathbb {Z}^l there exist an integer n n and a vector u ∈ Z l u\in \mathbb {Z}^l such that u + p i ( n ) v i ∈ S u+p_i(n)v_i\in S for each i ≤ k i\le k .