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

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 Zˡ be a set of positive upper Banach density, let p 1 (n), …, p k (n) p₁ (n), , pₖ (n) be polynomials with rational coefficients taking integer values on the integers and satisfying p i (0) = 0 pᵢ (0) =0, i = 1, …, k ; i=1, , k; then for any v 1, …, v k ∈ Z l v₁, , vₖ Zˡ there exist an integer n n and a vector u ∈ Z l u Zˡ such that u + p i (n) v i ∈ S u+pᵢ (n) vᵢ S for each <m