Polynomial Configurations on Integer Subsets with Positive Density

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized it, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of this polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We give a multiset extension of the polynomial theorem and derive several combinatorial consequences, including lower bounds for the size of the intersection and an “anti-Ramsey” rainbow result. We also prove a structure theorem for the polynomial multicorrelation sequences ∫