Multiple ergodic averages for three polynomials and applications

摘要

We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {l(1)p, l(2)p,..., l(k)p}. We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemeredi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all epsilon > 0 and every subset of the integers. the set {n is an element of N : d* (Lambda boolean AND (Lambda + p(1)(n)) boolean AND (Lambda + p(2)(n)) boolean AND (Lambda + p(3)(n))) > (d* (Lambda))(4) - epsilon} has bounded gaps for "most" choices of integer polynomials p(1), p(2), p(3).