Properties of multicorrelation sequences and large returns under some ergodicity assumptions

Abstract

We prove that given a measure preserving system (X, B, µ, T 1 ,. .. , T d) with commuting, ergodic transformations T i such that T i T −1 j are ergodic for all i = j, the multicorrelation sequence a(n) = X f 0 •T n 1 f 1 •.. .•T n d f d dµ can be decomposed as a(n) = ast(n) + aer(n), where ast is a uniform limit of d-step nilsequences and aer is a nullsequence (that is, lim N −M →∞

Decomposition of multicorrelation sequences and joint ergodicity

“…can be decomposed as a sum of a uniform limit of k-step nilsequences plus a nullsequence. For more general expressions (as in (3)), exploiting results from [18], it is also shown in [8] that, if we further assume ergodicity in all directions, i.e., T a 1 1 • . .…”

“…A partial answer towards this direction was obtained in [8] by the second author. Namely, [8,Theorem 1.5] shows that for any system (X, B, µ, T 1 , . .…”

“…can be decomposed as a sum of a uniform limit of D-step nilsequences plus a nullsequence. 8 The proof of this result makes essential use of a seminorm bound estimate obtained in [18], where the (infinitely many) ergodicity assumptions are reflected (see [8,Theorem 1.6]).…”

“…In [6], the first, third and fourth authors improved the seminorm bound estimates of [18] by imposing only finitely many ergodic assumptions. Although the results in [6] are stronger than those in [18], one cannot apply them directly to [8] to improve the aforementioned results, due to the incompatibility of the methods between [6] and [8] (see Subsection 2.3 for more details).…”

“…, p k (i.e., p i , p i − p j are non-constant for all 1 ≤ i, j ≤ k, i = j) 10 One can think of this last result as a strong independence property of the sequences (T p i (n) ) n∈Z , 1 ≤ i ≤ k in the weakly mixing case. It is reasonable to expect a characterization in the case of polynomial iterates, which naturally leads to a general notion of joint ergodicity: 8 Here D depends on k, d and the maximum degree of the pi's. It also has a connection to the number of van der Corput operations we have to run in the induction (see Remark 5.8 for details).…”