PROPERTIES OF MULTICORRELATION SEQUENCES AND LARGE RETURNS UNDER SOME ERGODICITY ASSUMPTIONS

摘要

We prove that given a measure preserving system (X, B, μ, T_1,…, T_d) with commuting, ergodic transformations T_i such that T_iT_j~(-1) are ergodic for all i ≠ j, the multicorrelation sequence a(n) = ∫_x f_o·T_1~n f_1·…·T_d~n f_d μ can be decomposed as a(n) = a_(st)(n)+a_(er)(n), where a_(st) is a uniform limit of d-step nilsequences and a_(er) is a nullsequence (that is, lim_(N-M)→∞1/(N-M)∑_(n=M)~(N-1)｜a_(er)｜~2 = 0). Under some additional ergodicity conditions onT_1,…,T_d we also establish a similar decomposition for polynomial multicorrelation sequences of the form a(n) = ∫_x f_0·∏_(i=1)~d T_i~(pi,1(n))f_1·…∏_(i=1)~d T_i~(pi,k(n)) f_k dμ, where each p_(i,k) : Z→Z is a polynomial map. We also show, for d = 2, that if T_1, T_2, T_1T_2~(-1) are invertible and ergodic, we have large triple intersections: for all ε > 0 and all A∈ B, the set {n∈Z: μ(A∩T_1~(-n) A∩T_2~(-n) A) ＞μ(A)~3 -ε} is syndetic. Moreover, we show that if T_1, T_2,T_1T_2~(-1) are totally ergodic, and we denote by p_n the n-th prime, the set {n ∈ N :μ(A∩T_1~(-(P_n)-1)A∩T_2~(-(p_n)-1)A) ＞ μ(A)~3-ε｝ has positive lower density.