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⊆ Z^l be a set of positive upper Banach density, let p 1 ( n ) , … , p k ( n ) p_1(n), ,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, ,k; then for any v 1 , … , v k ∈ Z l v_1, ,v_k∈ Z^l there exist an integer n n and a vector u ∈ Z l u∈ Z^l such that u + p i ( n ) v i ∈ S u+p_i(n)v_i∈ S for each <m