Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory (Q926375)

After introducing two notions of complexity of a system of polynomials \(p_1,\dots,p_r\in\mathbb{Z}[n]\), the authors characterize the limits of the expressions of the form \(\mu(A_0\cap T^{-p_1(n)}A_1\cap\dots\cap T^{-p_r(n)}A_r)\), where \(T\) is a skew-product transformation of a torus \(\mathbb{T}^d\) and \(A_i\subseteq\mathbb{T}^d\) are measurable sets. Using the dynamical results they obtain, they are able to build subsets of integers with specific combinatorial properties related with the polynomial Szemerédi theorem.