Joint ergodicity of sequences

Decomposition of multicorrelation sequences and joint ergodicity

“…, [p k (n)] ([•] denotes the floor function), as well (see [21]). These last two results also follow by a recent work of Frantzikinakis, [11], in which, for single T , we have a plethora of joint ergodicity results for a number of classes of iterates (not just polynomial). Finally, for real variable polynomial iterates, one is referred to [26].)…”

“…Remark Next, we provide the proof of Theorem 2.9. To this end, we recall a definition from [4] which is adapted from [11].…”

“…For multicorrelation sequences with general polynomial iterates, this D can be taken to be equal to the number of vdC operations we have to perform for all the iterates to become constant (e.g. [10] via [13], and [25] via [10]). At this point, a word of caution is necessary for the approach of this paper.…”

On arithmetic functions orthogonal to deterministic sequences

“…The system px, yq Þ Ñ px, y `xq is not a Furstenberg system of the Liouville function (see also slide no 6 in [17]). We recall that in [42] (see Theorem 1.3 therein), it is proved that In view of Corollary 2.11 (see also (23)) and Theorem A, in order to obtain u K C DISPec it is sufficient and necessary to have u K L 2 pKpI κ qq for each Furstenberg system κ of u.…”

“…Remark 5.9. In fact, already Frantzikinakis [17] (see slide no 10) showed that if u is generic then }u} u 2 " 0 if and only if…”