Ergodic averages for independent polynomials and applications

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper den-sity contains arbitrarily long arithmetic progressions. Bergelson and Leibman general-ized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in com-binatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation