Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory

We introduce two notions of complexity of a system of polynomials p(1),..., p(r) is an element of Z [n] and apply them to characterize the limits of the expressions of the form mu(A(0) boolean AND T-p1 ((n)) A(1) boolean AND...boolean AND T-pr ((n)) A(r) )where T is a skew-product transformation of a torus T-d and A(i) subset of T-d are measurable sets. The dynamical results obtained allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemeredi theorem.