Multiple ergodic averages for three polynomials and applications

Abstract

Abstract. 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 Szemerédi 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 ε > 0 and every subset of the integers Λ the set

Ergodic averages of commuting transformations with distinct degree polynomial iterates

“…If = 2, d 1 = d 2 = 1, and the joint action of the transformations T 1 , T 2 is ergodic, then the result remains true up to a change of the exponent on the right-hand side [10]. But even under similar ergodicity assumptions, the result probably fails when three exponents agree no matter what exponent one uses on the right-hand side (a conditional counterexample appears in [14,Proposition 5.2]).…”

Optimal lower bounds for multiple recurrence

“…Suppose (a 1 , a 2 , a 3 , a 4 ) = (0, 2, 3, 4). In this case V is the Q-span of (1, 1, 1, 1), (0, 2, 3, 4) and (0, 4,9,16). It follows that It follows from the Cauchy-Schwarz inequality that…”

“…In most cases where recurrence has been established, optimal recurrence can be obtained for weakly mixing systems (see [2] when the f i are polynomials and [3,11,12] for more general f i ), or when the functions are "independent" (see [15,16] for the case of linearly independent polynomials and [12] for more general f i with different growth). In the general case, besides the aforementioned paper [4], the main progress was obtained by Frantzikinakis in [9], where the case k ≤ 3 and the f i are polynomials is studied in detail.…”

Simultaneous polynomial recurrence

“…Bergelson, Host and Kra established in [5] that (1) does hold, under this additional assumption that T is ergodic in the case of (dependent) linear polynomials when ℓ = 2, 3. Frantzikinakis [7] has investigated the situation for higher degree polynomials.…”