1 Introduction
Let $G=(G,+)$ be a countable discrete abelian group and let $(T_g)_{g\in G}$ be a measurepreserving Gaction on a separable probability space $(X,\mathcal A,\mu )$ . We will call the quadruple $(X,\mathcal A,\mu , (T_g)_{g\in G})$ a measurepreserving system. A measurepreserving system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ is strongly mixing (or 2mixing) if for any $A_0,A_1\in \mathcal A$ , one has
The goal of this paper is to obtain new results about higherorder mixing properties of strongly mixing actions of abelian groups. These results are motivated by the following classical problem going back to Rohlin (who formulated it for ${\mathbb {Z}}$ actions; see [Reference Rohlin27]).
Rohlin’s Problem. Assume that a measurepreserving system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ is strongly mixing. Is it true that, given any $\ell \geq 2$ , the system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ is $(\ell +1)$ mixing? This would mean that for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any sequences $(g^{(1)}_k)_{k\in {\mathbb {N}}},\ldots,$ $(g^{(\ell )}_k)_{k\in {\mathbb {N}}}$ in G satisfying that,

(i) for any $j\in \{1,\ldots ,\ell \}$ ,
(1.2) $$ \begin{align} \lim_{k\rightarrow\infty}g^{(j)}_k=\infty, \end{align} $$ 
(ii) and, for any distinct $i,j\in \{1,\ldots ,\ell \}$ ,
(1.3) $$ \begin{align} \lim_{k\rightarrow\infty}(g^{(j)}_kg^{(i)}_k)=\infty, \end{align} $$
one has
While for ${\mathbb {Z}}$ actions Rohlin’s problem is still unsolved, an example for ${\mathbb {Z}}^2$ actions, due to Ledrappier, shows that, in general, mixing does not imply mixing of higher orders [Reference Ledrappier22] (the reader is referred to [Reference Schmidt30] for more Ledrappiertype examples for ${\mathbb {Z}}^d$ actions). More precisely, Ledrappier provided an example of a pair $S,T$ of commuting mixing automorphisms of a compact abelian group X such that, for some measurable set $A\subseteq X$ ,
where $\mu $ is the normalized Haar measure on X. The analysis of Ledrappier’s example undertaken in [Reference ArenasCarmona, Berend and Bergelson1] reveals that Ledrappier’s system is ‘almost mixing of all orders’ in the sense that, for any $\ell \in {\mathbb {N}}$ , if the sequences $(g_k^{(1)})_{k\in {\mathbb {N}}},\ldots,$ $(g_k^{(\ell )})_{k\in {\mathbb {N}}}$ in ${\mathbb {Z}}^2$ satisfy (1.2) and (1.3) and, in addition, the $\ell $ tuples $(g_k^{(1)},\ldots ,g_k^{(\ell )})$ avoid certain rather rarefied subsets of ${\mathbb {Z}}^{2\ell }$ , equation (1.4) holds for any measurable $A_0,\ldots ,A_{\ell }\subseteq X$ (see [Reference ArenasCarmona, Berend and Bergelson1, Theorem 3.3]). The results obtained in [Reference ArenasCarmona, Berend and Bergelson1] were extended in [Reference ArenasCarmona, Berend and Bergelson2] to a rather large family of systems of algebraic origin. The notable classes of ${\mathbb {Z}}$ actions for which it is known that 2mixing implies mixing of all orders include ergodic automorphisms of compact groups [Reference Rohlin27], mixing transformations with singular spectrum [Reference Host18], and mixing actions of finite rank [Reference Kalikow19, Reference Ryzhikov28]. It is also known that some natural actions of various locally compact groups possess the property of mixing of all orders (see, for example, [Reference Fayad and Kanigowski12, Reference Marcus24, Reference Mozes26, Reference Ryzhikov29]).
In view of the results obtained in [Reference ArenasCarmona, Berend and Bergelson1, Reference ArenasCarmona, Berend and Bergelson2], one might wonder if it could possibly be true that, similarly to the case of Ledrappier’s system, any strongly mixing action $(X,\mathcal A,\mu , (T_g)_{g\in G})$ of an abelian group G is, in some sense, almost mixing of all orders. The goal of this paper is to establish a result that can be interpreted as a positive answer to this question.
At this point, we would like to mention that in the special case when $G={\mathbb {Z}}$ , our main theorem (Theorem 1.21 below) has corollaries (Theorem 1.4 and Corollary 1.12) which provide new nontrivial characterizations of the notion of strong mixing in terms of the largeness of sets of the form
and
So, if it turns out that sets of the form (1.5) and (1.6) are not always cofinite, our results still imply that these sets are large in some natural sense, thereby establishing the validity of the claim that strongly mixing ${\mathbb {Z}}$ actions are almost mixing of all orders.
Let $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ be a measurepreserving system. Let $\ell \in {\mathbb {N}}$ and $\epsilon>0$ . For any $A_0,\ldots ,A_{\ell }\in \mathcal A$ consider the set
Clearly, the higher is the degree of multiple mixing of the system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ , the more massive should the set $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ be as a subset of $G^{\ell }$ . While, for $\ell =1$ , the strong mixing property of $(X,\mathcal A,\mu , (T_g)_{g\in G})$ implies that the set $R_{\epsilon }(A_0,A_1)$ is cofinite, this is no longer the case for $\ell \geq 2$ even if our system $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ is mixing of all orders. For example, for any 3mixing system and any $A\in \mathcal A$ with $\mu (A)\in (0,1)$ , one has that, if $\epsilon>0$ is small enough, the set
can only have a finite intersection with any of the ‘lines’ $\{(g,g)\,\,g\in G\}$ , $\{(g,0)\,\,g\in G\}$ and $\{(0,g)\,\,g\in G\}$ .
In what follows we will show that, for any mixing system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ , the subsets of $G^{\ell }$ which are of the form $\mathcal R_{\epsilon }(A_0,\ldots ,A_{\ell })$ possess a strong ubiquity property which we will call $\tilde {\Sigma }_{\ell }^*$ and which is quite a bit stronger than the properties of largeness associated with weakly and mildly mixing systems. In other words, we will show that for any strongly mixing system the complement of any set of the form $R_{\epsilon } (A_0,\ldots ,A_{\ell })$ is very ‘small’, giving meaning to the claim that $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ is ‘almost strongly mixing’ of all orders. This will be achieved with the help of $\mathcal R$ limits, a notion of convergence which is based on a classical combinatorial result due to Ramsey and, as we will see, is adequate for dealing with strongly mixing systems. (In particular, we will show that the $\tilde {\Sigma }_{\ell }^*$ property of the sets $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ implies the strong mixing of $(X,\mathcal A,\mu ,(T_g)_{g\in G}$ ).)
We would like to remark that while the results that we obtain are not as sharp as those obtained in [Reference ArenasCarmona, Berend and Bergelson1, Reference ArenasCarmona, Berend and Bergelson2], they have the advantage of being applicable to any strongly mixing system $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ , where G is a countable abelian group. Moreover, as will be demonstrated in §6, the versatility of $\mathcal R$ limits allows one to obtain new and recover some old results pertaining to multiple recurrence properties of weakly and mildly mixing actions of countable abelian groups. We would also like to mention that, as will be seen in §3, the utilization of $\mathcal R$ limits brings to life many new equivalent characterizations of strong mixing (some of which bear a strong analogy with the familiar characterizations of weak mixing via convergence in density and mild mixing via IPconvergence).
Before introducing the abovementioned notion of largeness for subsets of $G^{\ell }$ , we define a related and somewhat simpler notion in G.
Definition 1.1. Let $m\in {\mathbb {N}}$ , let $(G,+)$ be a countable abelian group, and let $E\subseteq G$ .

(1) We say that E is a $\Sigma _m$ set if it is of the form
$$ \begin{align*}\{g_{k_1}^{(1)}+\cdots+g_{k_m}^{(m)}\,\,k_1<\cdots<k_m\}\end{align*} $$where, for each $j\in \{1,\ldots ,m\}$ , $(g_k^{(j)})_{k\in {\mathbb {N}}}$ is a sequence in G which satisfies $\lim _{k\rightarrow \infty }g_k^{(j)}=\infty $ . 
(2) We say that E is a $\Sigma _m^*$ set if it has a nontrivial intersection with every $\Sigma _m$ set.
Remark 1.2

(a) Note that a subset of G is $\Sigma _1^*$ if and only if it is cofinite. On the other hand, for any $m\geq 2$ , a $\Sigma _m^*$ set does not need to be cofinite. Moreover, one can show that for each $m\geq 2$ , there exists a $\Sigma _{m}^*$ set which fails to be a $\Sigma _{n}^*$ set for each $n<m$ [Reference Bergelson and Zelada8].

(b) The notion of $\Sigma _m^*$ is similar to (but much stronger than) the notion of IP $^*$ which has an intrinsic connection to mild mixing and which plays an instrumental role in IP ergodic theory and in Ramsey theory (see, for example, [Reference Bergelson and McCutcheon5, Reference Furstenberg14, Reference Furstenberg and Katznelson15]). The connection between these two notions will be discussed in detail in §5.
Since the sets $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ are, by definition, subsets of $G^{\ell }$ , the abovedefined notion of $\Sigma _m^*$ has to be ‘upgraded’ to the subsets of the Cartesian power $G^{\ell }$ in order to be useful in the study of the asymptotic behavior of multiparameter expressions of the form
However, it is worth noting that the family of $\Sigma _m^*$ sets is quite adequate for dealing with ‘diagonal’ multicorrelation sequences. In the case $G={\mathbb {Z}}$ , such diagonal sequences have the form
where $a_1,\ldots ,a_{\ell }\in {\mathbb {Z}}$ , and play an instrumental role in Furstenberg’s ergodic approach to Szemerédi’s theorem [Reference Furstenberg13, Reference Furstenberg14]. For example, our main result (Theorem 1.21), while dealing with the multiparameter expressions (1.8), has strong corollaries of a ‘diagonal’ nature. The following theorem (which is a version of Theorem 4.4 below) is an example of a new result of this kind. Note the appearance of $\Sigma _{\ell }^*$ sets in the formulation.
Theorem 1.3. Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ be a strongly mixing system, and let the homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be such that, for any $j\in \{1,\ldots ,\ell \}$ , $\ker (\phi _j)$ is finite and, for any $i\neq j$ , $\ker (\phi _j\phi _i)$ is also finite. Then, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set
is $\Sigma _{\ell }^*$ .
When G is finitely generated, Theorem 1.3 has a stronger version (Theorem 4.2), which in the case $G={\mathbb {Z}}$ can be formulated as follows.
Theorem 1.4. Let $(X,\mathcal A,\mu , T)$ be a measurepreserving system, let $\ell \in {\mathbb {N}}$ , and let $a_1,\ldots ,a_{\ell }$ be distinct nonzero integers. Then T is strongly mixing if and only if, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set
is $\Sigma _{\ell }^*$ .
For a related result see [Reference Bergelson and Zelada7, Theorem 1.11]. See also [Reference Kuang and Ye20].
Remark 1.5. One can view Theorem 1.4 as a strongly mixing analogue of two theorems due to Furstenberg which pertain to weak and mild mixing (see Theorems 4.11 and 9.27 in [Reference Furstenberg14], respectively). The first of these two theorems states that the assumption that $(X,\mathcal A,\mu , T)$ is weakly mixing implies (and is implied by the fact) that the sets $R^{a_1,\ldots ,a_{\ell }}_{\epsilon }(A_0,\ldots ,A_{\ell })$ defined in (1.11) have uniform density one. The second one states that the assumption that $(X,\mathcal A,\mu ,T)$ is mildly mixing implies (and is implied by) the IP $^*$ property of the sets $R^{a_1,\ldots ,a_{\ell }}_{\epsilon }(A_0,\ldots ,A_{\ell })$ . These theorems are instrumental for the proofs of the ergodic Szemerédi [Reference Furstenberg13] and IPSzemerédi [Reference Furstenberg and Katznelson15] theorems.
Note that, for $\ell =1$ , both diagonal (see (1.9)) and multiparameter (see (1.8)) multicorrelation sequences reduce to the classical expression $\mu (A_0\cap T_{g}A_1)$ . The following theorem (which is a very special case of stronger results to be established in this paper) shows that, even in the rather degenerated case $\ell =1$ , $\Sigma _m^*$ sets provide a new characterization for the notion of strong mixing for actions of abelian groups.
Theorem 1.6. Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system. The following statements are equivalent.

(i) $(T_g)_{g\in G}$ is strongly mixing. In other words, for any $\epsilon>0$ and any $A_0,A_1\in \mathcal A$ , the set
$$ \begin{align*}R_{\epsilon}(A_0,A_1)=\{g\in G\,\,\mu(A_0\cap T_g A_1)\mu(A_0)\mu(A_1)<\epsilon\}\end{align*} $$is cofinite (that is, it is $\Sigma _1^*$ in G). 
(ii) For any $m\in {\mathbb {N}}$ , any $\epsilon>0$ and any $A_0,A_1\in \mathcal A$ , the set $R_{\epsilon }(A_0,A_1)$ is $\Sigma _m^*$ in G.

(iii) There exists an $m\in {\mathbb {N}}$ such that, for any $\epsilon>0$ and any $A_0,A_1\in \mathcal A$ , the set $R_{\epsilon }(A_0,A_1)$ is $\Sigma _m^*$ in G.
We next define the modified versions of $\Sigma _m$ and $\Sigma _m^*$ sets which will be instrumental in dealing with the multiple mixing properties of strongly mixing systems.
Definition 1.7. Let $(G,+)$ be a countable abelian group and let $(g_k)_{k\in {\mathbb {N}}}$ and $(h_k)_{k\in {\mathbb {N}}}$ be two sequences in G. We say that $(g_k)_{k\in {\mathbb {N}}}$ and $(h_k)_{k\in {\mathbb {N}}}$ grow apart if $\lim _{k\rightarrow \infty } (g_kh_k)=\infty $ .
Definition 1.8. Let $(G,+)$ be a countable abelian group, let $d\in {\mathbb {N}}$ and let $(\textbf g_k)_{k\in {\mathbb {N}}}=(g_{k,1},\ldots ,g_{k,d})_{k\in {\mathbb {N}}}$ be a sequence in $G^d$ . We say that $(\textbf g_k)_{k\in {\mathbb {N}}}$ is nondegenerated if, for each $j\in \{1,\ldots ,d\}$ ,
Definition 1.9. Let $d,m\in {\mathbb {N}}$ and let $(G,+)$ be a countable abelian group.

(1) We say that $E\subseteq G^d$ is a $\tilde {\Sigma }_m$ set if it is of the form
$$ \begin{align*}\{\textbf g_{k_1}^{(1)}+\cdots+\textbf g_{k_m}^{(m)}\,\,k_1<\cdots<k_m\}\end{align*} $$where, for each $j\in \{1,\ldots ,m\}$ , $(\textbf g_k^{(j)})_{k\in {\mathbb {N}}}=(g_{k,1}^{(j)},\ldots ,g_{k,d}^{(j)})_{k\in {\mathbb {N}}}$ is a non degenerated sequence in $G^d$ and for any distinct $t,t'\in \{1,\ldots ,d\}$ the sequences $(g_{k,t}^{(j)})_{k\in {\mathbb {N}}}$ and $(g_{k,t'}^{(j)})_{k\in {\mathbb {N}}}$ grow apart. (Note that if $d=1$ , then $E\subseteq G$ is a $\Sigma _m$ set if and only if it is a $\tilde {\Sigma }_m$ set.) 
(2) We say that $E\subseteq G^d$ is a $\tilde \Sigma _m^*$ set if it has a nontrivial intersection with every $\tilde \Sigma _m$ set in $G^d$ .
Remark 1.10. The main difference between $\tilde {\Sigma }_m$ sets and $\Sigma _m$ sets is that $\tilde {\Sigma }_m$ sets are subsets of Cartesian powers of G and have a builtin feature which guarantees that, asymptotically, the elements of $\tilde {\Sigma }_m$ sets stay away from ‘degenerated’ subsets such as the following subsets of $G^3$ : $\{(g,g,g)\,\,g\in G\}$ , $\{(g,2g,0)\,\,g\in G\}$ and $\{(g,g,h)\,\,g, h\in G\}$ .
The following theorem, which is a corollary of Theorem 1.21 below, demonstrates the relevance of $\tilde {\Sigma }_m$ sets for dealing with mixing of higher orders.
Theorem 1.11. Let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system. The following statements are equivalent.

(i) $(T_g)_{g\in G}$ is strongly mixing.

(ii) For any $\ell \in {\mathbb {N}}$ , any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set
$$ \begin{align*}&R_{\epsilon}(A_0,\ldots,A_{\ell})\\ &\quad=\bigg\{(g_1,\ldots,g_{\ell})\in G^{\ell}\,\bigg\,\,\bigg\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_{\ell}}A_{\ell})\prod_{j=0}^{\ell} \mu( A_j)\bigg<\epsilon\bigg\} \end{align*} $$is $\tilde {\Sigma }_{\ell }^*$ in $G^{\ell }$ . 
(iii) There exists an $\ell \in {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ is $\tilde {\Sigma }_{\ell }^*$ in $G^{\ell }$ .
We take the liberty of stating explicitly the following special case of Theorem 1.11 to stress the applicability of the apparatus developed in this paper to ${\mathbb {Z}}$ actions.
Corollary 1.12. Let $(X,\mathcal A,\mu , T)$ be a measurepreserving system. The following statements are equivalent.

(i) T is strongly mixing.

(ii) For any $\ell \in {\mathbb {N}}$ , any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set
$$ \begin{align*}&R_{\epsilon}(A_0,\ldots,A_{\ell})\\ &\quad=\bigg\{(n_1,\ldots,n_{\ell})\in {\mathbb{Z}}^{\ell}\,\bigg\,\,\bigg\mu(A_0\cap T^{n_1}A_1\cap\cdots \cap T^{n_{\ell}}A_{\ell})\prod_{j=0}^{\ell} \mu( A_j)\bigg<\epsilon\bigg\}\end{align*} $$is $\tilde {\Sigma }_{\ell }^*$ in ${\mathbb {Z}}^{\ell }$ . 
(iii) There exists an $\ell \in {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ is $\tilde {\Sigma }_{\ell }^*$ in ${\mathbb {Z}}^{\ell }$ .
We introduce now the notion of convergence that is utilized in the proof of Theorem 1.11 and is based on the classical Ramsey theorem (which, for convenience of the reader, we state below). We remark that variants of this notion of convergence can also be found in [Reference Bojańczyk, Kopczyński and Toruńczyk10, Reference Campbell and McCutcheon11, Reference Kubiś and Szeptycki21, Reference Lorentz23, Reference McCutcheon25, Reference Sucheston31]. Given $m\in {\mathbb {N}}$ and an infinite set $S\subseteq {\mathbb {N}}$ , we denote by $S^{(m)}$ the family of all melement subsets of S. When writing $\{k_1,\ldots ,k_m\}\in S^{(m)}$ , we will always assume that $k_1<\cdots <k_m$ .
Theorem 1.13. (Ramsey’s theorem)
Let $r,m\in {\mathbb {N}}$ and let $C_1,\ldots ,C_r\subseteq {\mathbb {N}}^{(m)}$ be such that
Then there exist $j_0\in \{1,\ldots ,r\}$ and an infinite subset $S\subseteq {\mathbb {N}}$ satisfying $S^{(m)}\subseteq C_{j_0}$ .
Remark 1.14. It is easy to see that Theorem 1.13 can be formulated in the following equivalent form that will be frequently used in the sequel.
Let $r,m\in {\mathbb {N}}$ , let P be an infinite subset of ${\mathbb {N}}$ and let $C_1,\ldots ,C_r\subseteq {\mathbb {N}}^{(m)}$ be such that
Then there exist $j_0\in \{1,\ldots ,r\}$ and an infinite subset $S\subseteq P$ satisfying $S^{(m)}\subseteq ~C_{j_0}$ .
Definition 1.15. Let $m\in {\mathbb {N}}$ , let $(X,d)$ be a compact metric space, let $x\in X$ , let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ be an ${{\mathbb {N}}^{(m)}\text {sequence}}$ in X, and let S be an infinite subset of ${\mathbb {N}}$ . We write
if, for every $\epsilon>0$ , there exists $\alpha _0\in {\mathbb {N}}^{(m)}$ such that, for any $\alpha \in S^{(m)}$ satisfying ${\min \alpha>\max \alpha _0}$ , one has
The following theorem can be viewed as a version of Bolzano–Weierstrass theorem for ${\mathcal R\text {convergence}}$ . It follows from Theorem 1.13 with the help of a diagonalization argument.
Theorem 1.16. Let $m\in {\mathbb {N}}$ , let $(X,d)$ be a compact metric space and let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ be an ${\mathbb {N}}^{(m)}$ sequence in X. Then, for any infinite set $S_1\subseteq {\mathbb {N}}$ , there exist an $x\in X$ and an infinite set $S\subseteq S_1$ such that
Remark 1.17. Let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ be an ${\mathbb {N}}^{(m)}$ sequence in a compact metric space $(X,d)$ . The aboveintroduced $\mathcal R$ limits have an intrinsic connection with iterated limits of the form
The goal of this extended remark is to clarify this connection.

(a) Using the compactness of X, one can show with the help of a diagonalization argument that for any increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ , there exists a subsequence $(k_j')_{j\in {\mathbb {N}}}$ for which all the limits in (1.16) exist.

(b) By Theorem 1.16, there exists an increasing sequence of natural numbers $(k_j)_{j\in {\mathbb {N}}}$ so that, for $S=\{k_j\,\,j\in {\mathbb {N}}\}$ , ${\mathop { \mathcal {R}{\text {}\mathrm {lim}}}_{{\alpha \in S^{(m)}}}} x_{\alpha }$ exists. Let $(k_j')_{j\in {\mathbb {N}}}$ be the subsequence of $(k_j)_{j\in {\mathbb {N}}}$ which is guaranteed to exist by (a). Letting $S_1=\{k_j'\,\,j\in {\mathbb {N}}\}$ , we have
(1.17) $$ \begin{align} {\mathop { \mathcal {R}{\text{}\mathrm{lim}}}_{{\alpha\in S_1^{(m)}}}}\ x_{\alpha}=\lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k^{\prime}_{j_1},\ldots,k^{\prime}_{j_m}\}}. \end{align} $$ 
(c) When $X=\{1,\ldots ,r\}$ , one can use (a) to prove Theorem 1.13. Let $r,m\in {\mathbb {N}}$ and consider a partition ${\mathbb {N}}^{(m)}=\bigcup _{j=1}^r C_j$ . Let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ be defined by $x_{\alpha }=j$ if $\alpha \in C_j$ . For some increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in ${\mathbb {N}}$ there exists a $j_0\in \{1,\ldots ,r\}$ such that
$$ \begin{align*} \lim_{j_1\rightarrow\infty}\cdots\lim_{j_m\rightarrow\infty}x_{\{k_{j_1},\ldots,k_{j_m}\}}=j_0. \end{align*} $$By using a diagonalization argument, we obtain a subsequence $(k_j')_{j\in {\mathbb {N}}}$ of $(k_j)_{j\in {\mathbb {N}}}$ with the property that $x_{\{k_{j_1}',\ldots ,k_{j_m}'\}}=j_0$ for any $j_1<\cdots <j_m$ . Now let $S=\{k_j'\,\, j\in {\mathbb {N}}\}$ . It follows that $S^{(m)}\subseteq C_{j_0}$ .
Before formulating our main result, we need two more definitions.
Definition 1.18. Let $m\in {\mathbb {N}}$ and let $(G,+)$ be a countable abelian group. For any sequence $(\textbf g_k)_{k\in {\mathbb {N}}}=(g_{k,1},\ldots ,g_{k,m})_{k\in {\mathbb {N}}}$ and any $\alpha =\{k_1,\ldots ,k_m\}\in {\mathbb {N}}^{(m)}$ we let
where $k_1<\cdots <k_m$ .
Definition 1.19. Let $m\in {\mathbb {N}}$ , let $(G,+)$ be a countable abelian group and let
be sequences in $G^m$ . We say that $(\textbf g_k)_{k\in {\mathbb {N}}}$ and $(\textbf h_k)_{k\in {\mathbb {N}}}$ are essentially distinct if, for each $t\in \{1,\ldots ,m\}$ , $(g_{k,t})_{k\in {\mathbb {N}}}$ and $(h_{k,t})_{k\in {\mathbb {N}}}$ grow apart (that is, $\lim _{k\rightarrow \infty }(g_{k,t}h_{k,t})=\infty $ ).
Remark 1.20. The following observation indicates the natural connection between nondegenerated, essentially distinct sequences in $G^m$ and $\tilde {\Sigma }_m$ sets. Let $d,m\in {\mathbb {N}}$ and let $(G,+)$ be a countable abelian group. Then for any nondegenerated and essentially distinct sequences
in $G^m$ , the set
is a $\tilde {\Sigma }_m$ set in $G^d$ .
We are now ready to formulate our main result (it appears as Theorem 3.1 in §3). It incorporates some of the characterizations of strongly mixing systems which were mentioned above.
Theorem 1.21. Let $\ell \in {\mathbb {N}}$ , let $(G,+)$ be a countable abelian group and let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system. The following statements are equivalent.

(i) $(T_g)_{g\in G}$ is strongly mixing.

(ii) For any nondegenerated and essentially distinct sequences
$$ \begin{align*}(\textbf g_k^{(j)})_{k\in{\mathbb{N}}}, \quad j\in\{1,\ldots,\ell\},\end{align*} $$in $G^{(\ell )}$ , there exists an infinite $S\subseteq {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ ,(1.19) $$ \begin{align} {\mathop { \mathcal {R}{\text{}\mathrm{lim}}}_{{\alpha\in S^{(\ell)}}}}\, \mu(A_0\cap T_{ g^{(1)}_{\alpha}}A_1\cap \cdots\cap T_{ g^{(\ell)}_{\alpha}}A_{\ell})=\prod_{j=0}^{\ell}\mu(A_j). \end{align} $$More explicitly, if
$$ \begin{align*} (\textbf g^{(j)}_k)_{k\in{\mathbb{N}}}=(g^{(j)}_{k,1},\ldots,g^{(j)}_{k,\ell})_{k\in{\mathbb{N}}}, \end{align*} $$for each $j\in \{1,\ldots ,\ell \},$ then$$ \begin{align*} {\mathop { \mathcal {R}{\text{}\mathrm{lim}}}_{{\{k_1,\ldots,k_{\ell}\}\in S^{(\ell)}}}} \mu(A_0\cap T_{ g_{k_1,1}^{(1)}+\cdots+ g_{k_{\ell},\ell}^{(1)}}A_1\cap\cdots\cap T_{ g_{k_1,1}^{(\ell)}+\cdots+ g_{k_{\ell},\ell}^{(\ell)}}A_{\ell})=\prod_{j=0}^{\ell} \mu(A_j). \end{align*} $$ 
(iii) For any $\epsilon>0$ and any $A_0,\ldots ,A_{\ell }\in \mathcal A$ , the set
$$ \begin{align*} &R_{\epsilon}(A_0,\ldots,A_{\ell})\\&\quad=\bigg\{(g_1,\ldots,g_{\ell})\in G^{\ell}\,\bigg\,\,\bigg\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_{\ell}}A_{\ell})\prod_{j=0}^{\ell} \mu( A_j)\bigg<\epsilon\bigg\} \end{align*} $$is $\tilde \Sigma _{\ell }^*$ in $G^{\ell }$ . 
(iv) For any $\epsilon>0$ and any $A_0,A_1\in \mathcal A$ , the set $R_{\epsilon }(A_0,A_1)$ is $\Sigma _{\ell }^*$ in G.
The structure of this paper is as follows. In §2 we review some basic facts about couplings of probability spaces and establish some auxiliary results which will be needed in §3 and §6. In §3 we prove our main result, Theorem 1.21 (=Theorem 3.1). In §4 we derive some diagonal results for strongly mixing systems. In §5 we describe the largeness properties of $\tilde {\Sigma }_m^*$ sets and, more specifically, of the sets $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ . We also juxtapose the properties of $\tilde {\Sigma }_m^*$ sets with those of $\tilde {{\text {IP}}}^{\mathrm {*}}$ sets and sets of uniform density one which are characteristic, correspondingly, of mild and weak mixing. In §6 we utilize the methods developed in §2 and §5 to obtain analogues of Theorem 1.21 for mildly and weakly mixing systems.
Remark 1.22. Throughout this paper, we will be tacitly assuming that the measurepreserving systems $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ that we are working with are regular, meaning that the underlying probability space $(X,\mathcal A,\mu )$ is regular (that is, X is a compact metric space and $\mathcal A=\text {Borel}(X)$ ). Note that this assumption can be made without loss of generality since every separable measurepreserving system is equivalent to a regular one (see, for instance, [Reference Furstenberg14, Proposition 5.3]).
2 Some auxiliary facts involving couplings and $\mathcal R$ limits
In this section we review some basic facts about couplings of probability spaces and establish some auxiliary results which will be needed in §3 and §6.
Definition 2.1. Let $N\in {\mathbb {N}}$ . Given regular probability spaces $\textbf X_j=(X_j,\mathcal A_j,\mu _j)$ , $j\in \{1,\ldots ,N\}$ , a coupling of $\textbf X_1,\ldots ,\textbf X_N$ is a Borel probability measure $\unicode{x3bb} $ defined on the measurable space
having the property that, for any $j\in \{1,\ldots ,N\}$ and any $A\in \mathcal A_j$ , $\unicode{x3bb} (\pi _j^{1}(A))=\mu _j(A)$ , where $\pi _j:\prod _{i=1}^NX_i\rightarrow X_j$ is the projection map onto the jth coordinate of $\prod _{j=1}^N X_j$ . (A coupling is just a joining of the trivial measurepreserving systems $(X_j,\mathcal A_j,\mu _j, \text {Id}_j)$ , $j\in \{1,\ldots ,N\}$ , where $\text {Id}_j:X_j\rightarrow X_j$ denotes the identity map on $X_j$ .)
We will let $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ denote the set of all couplings of $\textbf X_1,\ldots ,\textbf X_N$ . $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ is a closed subspace of the set of all probability Borel measures on $\prod _{j=1}^NX_j$ endowed with the ${\text {weak*}}$ topology. With this topology, $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ is a compact metrizable space. Given a sequence $(\unicode{x3bb} _k)_{k\in {\mathbb {N}}}$ in $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ ,
if and only if, for any $A_1\in \mathcal A_1,\ldots,$ $A_N\in \mathcal A_N$ ,
The following proposition follows immediately from the compactness of $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ and Theorem 1.16.
Proposition 2.2. Let $\textbf X_j=(X_j,\mathcal A_j,\mu _j)$ , $j\in \{1,\ldots ,N\}$ , be regular probability spaces. For any $m\in {\mathbb {N}}$ , any infinite $S\subseteq {\mathbb {N}}$ and any ${\mathbb {N}}^{(m)}$ sequence $(\unicode{x3bb} _{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ in $\mathcal C(\textbf X_1,\ldots ,\textbf X_N)$ ,
if and only if, for any $A_1\in \mathcal A_1,\ldots,$ $A_N\in \mathcal A_N$ ,
Our next goal is to establish a useful criterion for mixing of higher orders (Proposition 2.9). First, we need a definition and two lemmas.
Definition 2.3. Let $(Z,\mathcal D,\unicode{x3bb} )$ be a regular probability space and let, for each $k\in {\mathbb {N}}$ , $T_k:Z\rightarrow Z$ be a measurepreserving transformation. The sequence $(T_k)_{k\in {\mathbb {N}}}$ has the mixing property if, for every $A_0,A_1\in \mathcal D$ ,
Remark 2.4

(a) If each of the transformations $T_k$ , $k\in {\mathbb {N}}$ , is invertible, $(T_k)_{k\in {\mathbb {N}}}$ has the mixing property if and only if $(T_k^{1})_{k\in {\mathbb {N}}}$ has the mixing property.

(b) $(T_k)_{k\in {\mathbb {N}}}$ has the mixing property if and only if, for any $f,g\in L^2(\mu )$ ,
$$ \begin{align*}\lim_{k\rightarrow\infty}\int_X fT_kg\,\text{d}\mu=\int_Xf\,\text{d}\mu\int_Xg\,\text{d}\mu.\end{align*} $$
Lemma 2.5. Let $\textbf X=(X,\mathcal A,\mu )$ and $\textbf Y=(Y,\mathcal B,\nu )$ be regular probability spaces. For each $k\in {\mathbb {N}}$ , let $T_k:Y\rightarrow Y$ be a measurepreserving transformation, and assume that the sequence $(T_k)_{k\in {\mathbb {N}}}$ has the mixing property. Let $\unicode{x3bb} _0$ be a coupling of $\textbf X$ and $\textbf Y$ . Assume that $\unicode{x3bb} $ is a probability measure on $\mathcal A\otimes \mathcal B$ such that, for any $A\in \mathcal A$ and $B\in \mathcal B$ , one has
Then $\unicode{x3bb} =\mu \otimes \nu $ .
Proof. Note that it suffices to show that, for any $A\in \mathcal A$ and $B\in \mathcal B$ ,
Fix $A\in \mathcal A$ and $B\in \mathcal B$ . Since , we have by (2.1) that
Note that and, if we regard $\mathcal B$ as a sub $\sigma $ algebra of $\mathcal A\otimes \mathcal B$ , $\unicode{x3bb} _0_{\mathcal B}=\nu $ . The rightmost expression in (2.3) equals
where denotes the conditional expectation of with respect to $\mathcal B$ .
But $(T_k)_{k\in {\mathbb {N}}}$ has the mixing property, so the rightmost expression in (2.4) equals
By noting that
we have that (2.5) equals $\mu (A)\nu (B)$ .
Lemma 2.6. Let $m\in {\mathbb {N}}$ , let $(X,d)$ be a compact metric space, and let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m+1)}}$ be an ${\mathbb {N}}^{(m+1)}$ sequence in X. Assume that there exists an infinite $S\subseteq {\mathbb {N}}$ with the following properties: (a) for some $x\in X$ , ${\mathop { \mathcal {R}{\text {}\mathrm {lim}}}_{{\alpha \in S^{(m+1)}}}} x_{\alpha }=x$ ; (b) for each $k\in S$ , there exists $y_k\in X$ such that
Then
Proof. Let $\epsilon>0$ . Note that (1) there exists $k_0\in S$ such that, for any $\alpha \in S^{(m+1)}$ with $k_0\leq \min \alpha $ , $d(x_{\alpha }, x)<{\epsilon }/{2}$ and (2) for any $k\in S$ , there exists an $\alpha _k\in S^{(m)}$ such that, for any $\alpha \in S^{(m)}$ with $\min \alpha>\max (\alpha _k\cup \{k\})$ , $d(x_{\{k\}\cup \alpha },y_k)<{\epsilon }/{2}$ . It follows that, for any $k\in S$ with $k\geq k_0$ and any $\alpha \in S^{(m)}$ with $\min \alpha>\max (\alpha _k\cup \{k\})$ , $d(y_k,x)<d(x_{\{k\}\cup \alpha },y_k)+d(x_{\{k\}\cup \alpha },x)<\epsilon $ . Since $\epsilon>0$ was arbitrary,
Remark 2.7. Let $m\in {\mathbb {N}}$ and let $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m+1)}}$ be an ${\mathbb {N}}^{(m+1)}$ sequence in a compact metric space X. By applying Theorem 1.16 first to the ${\mathbb {N}}^{(m)}$ sequence $(\omega _{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}=((x_{\{k\}\cup \alpha })_{k\in {\mathbb {N}}})_{\alpha \in {\mathbb {N}}^{(m)}}$ in $X^{\mathbb {N}}$ (here $x_{\{k\}\cup \alpha }=x_0$ for some fixed $x_0\in X$ , whenever $k\geq \min \alpha $ ), and then to the ${\mathbb {N}}^{(m+1)}$ sequence $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m+1)}}$ , we obtain an infinite set $S\subseteq {\mathbb {N}}$ for which (a) and (b) in the statement of Lemma 2.6 hold. A similar reasoning shows that one can pick S to be a subset of any prescribed in advance infinite set $S_1\subseteq {\mathbb {N}}$ .
Remark 2.8. In Remark 1.17(c), we indicated how the utilization of iterated limits
leads to a proof of Ramsey’s theorem (Theorem 1.13). In this remark, we show that Lemma 2.6 and Remark 2.7 (which are corollaries of Ramsey’s Theorem) imply that, for any infinite set $S_1\subseteq {\mathbb {N}}$ and any ${\mathbb {N}}^{(m)}$ sequence $(x_{\alpha })_{\alpha \in {\mathbb {N}}^{(m)}}$ in a compact metric space X, there exists an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in $S_1$ such that, for $S=\{k_j\,\,j\in {\mathbb {N}}\}$ , each of the limits in the formula
exists. The proof is by induction on $m\in {\mathbb {N}}$ . When $m=1$ , the result follows from the compactness of X. Now let $m>1$ and let $S_1$ be an infinite subset of ${\mathbb {N}}$ . By Remark 2.7 and Lemma 2.6, there exists an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in $S_1$ such that, for $S=\{k_j\,\,j\in {\mathbb {N}}\}$ ,
The result now follows from the inductive hypothesis applied to the infinite set $S\subseteq {\mathbb {N}}$ and the ${\mathbb {N}}^{(m1)}$ sequence $((x_{\{k\}\cup \alpha })_{k\in {\mathbb {N}}})_{\alpha \in {\mathbb {N}}^{(m1)}}$ in the compact metric space $X^{\mathbb {N}}$ .
The following proposition provides a useful technical tool for establishing higherorder mixing properties of measurepreserving systems. It will be instrumental in §3 for dealing with strongly mixing systems and in §6 where we will focus on mildly and weakly mixing systems.
Proposition 2.9. Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system, let $\ell \in {\mathbb {N}}$ and, for each $j\in \{1,\ldots ,\ell \}$ , let
be a sequence in $G^{\ell }$ . Suppose that, for any $t\in \{1,\ldots ,\ell \}$ and any $j\in \{1,\ldots ,\ell \}$ , $(T_{g_{k,t}^{(j)}})_{k\in {\mathbb {N}}}$ has the mixing property and that, for any t and any $i\neq j$ , $(T_{(g_{k,t}^{(j)}g_{k,t}^{(i)})})_{k\in {\mathbb {N}}}$ also has the mixing property. Then, there exists an infinite set $S\subseteq {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ ,
Proof. The proof is by induction on $\ell $ . When $\ell =1$ , it follows from our hypothesis that, for any $A_0,A_1\in \mathcal A$ ,
Now fix $\ell \in {\mathbb {N}}$ and suppose that Proposition 2.9 holds for any $\ell '\leq \ell $ . Let $\textbf X=(X,\mathcal A,\mu )$ and let $\mu _{\Delta }\in \mathcal C=\mathcal C(\underbrace {\textbf X,\ldots , \textbf X}_{\ell +2\text { times}})$ be defined by $\mu (A_0\times \cdots \times A_{\ell +1})=\mu (A_0\cap \cdots \cap A_{\ell +1})$ . By the inductive hypothesis, there exists an infinite $S\subseteq {\mathbb {N}}$ such that, for any $A_1,\ldots ,A_{\ell +1}\in \mathcal A$ ,
By Theorem 1.16 and the compactness of $\mathcal C$ , there exist an infinite set $S_0\subseteq S$ and $\unicode{x3bb} _0\in \mathcal C$ such that, for any $A_0,\ldots ,A_{\ell +1}\in \mathcal A$ ,
Likewise, there exist an infinite set $S_1\subseteq S_0$ and $\unicode{x3bb} \in \mathcal C$ such that, for any $A_0,\ldots , A_{\ell +1}\in \mathcal A$ ,
Let $\textbf {Y}=(\prod _{j=1}^{\ell +1} X,\bigotimes _{j=1}^{\ell +1}\mathcal A, \bigotimes _{j=1}^{\ell +1}\mu )$ . Note that (2.6) holds if we substitute $S_1$ for S and (2.7) holds when we substitute $S_1$ for $S_0$ . Performing this substitution and applying first (2.7) and then (2.6) to $A_1,\ldots ,A_{\ell +1}\in \mathcal A$ , we have
Also, trivially, for any $A_0\in \mathcal A$ ,
Thus, $\unicode{x3bb} _0$ is a coupling of $\textbf X$ and $\textbf Y$ .
Using formula (2.7), Lemma 2.6 and applying (2.8) to the set $S_1=\{k_j\,\,j\in {\mathbb {N}}\}$ (where we assume that $(k_j)_{j\in {\mathbb {N}}}$ is an increasing sequence), we have
For each $j\in {\mathbb {N}}$ , let $\textbf T_j=T_{g_{k_j,1}^{(1)}}\times \cdots \times T_{g_{k_j,1}^{(\ell +1)}}$ . Note that, for any increasing sequence $(t_s)_{s\in {\mathbb {N}}}$ in ${\mathbb {N}}$ , there exist a subsequence $(t^{\prime }_s)_{s\in {\mathbb {N}}}$ and a measure $\unicode{x3bb} '\in \mathcal C(\textbf X,\textbf Y)$ , such that, for any $A\in \mathcal A$ and any $B\in \bigotimes _{j=1}^{\ell +1}\mathcal A$ , $\lim _{s\rightarrow \infty }\unicode{x3bb} _0(A\times \textbf T_{t^{\prime }_s}B)=\unicode{x3bb} '(A\times B)$ . By (2.9), $\unicode{x3bb} '=\unicode{x3bb} $ and hence, for any $A\in \mathcal A$ and any $B\in \bigotimes _{j=1}^{\ell +1}\mathcal A$ , $\lim _{j\rightarrow \infty }\unicode{x3bb} _0(A\times \textbf T_j B)=\unicode{x3bb} (A\times B)$ .
By Lemma 2.5 applied to $\textbf X=(X,\mathcal A,\mu )$ , $\textbf Y=(\prod _{j=1}^{\ell +1} X,\bigotimes _{j=1}^{\ell +1}\mathcal A, \bigotimes _{j=1}^{\ell +1}\mu )$ and the sequence of measurepreserving transformations $(T^{1}_{g_{k_j,1}^{(1)}}\times \cdots \times T^{1}_{g_{k_j,1}^{(\ell +1)}})_{j\in {\mathbb {N}}}$ , we have that $\unicode{x3bb} =\bigotimes _{j=0}^{\ell +1}\mu $ . It follows that, for any $A_0,\ldots ,A_{\ell +1}\in \mathcal A$ ,
completing the proof.
3 Strongly mixing systems are ‘almost’ strongly mixing of all orders
In this section we will prove the following theorem (Theorem 1.21 from the Introduction) which is the main result of this paper.
Theorem 3.1. Let $\ell \in {\mathbb {N}}$ and let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system. The following statements are equivalent.

(i) $(T_g)_{g\in G}$ is strongly mixing.

(ii) For any $\ell $ nondegenerated and essentially distinct sequences
$$ \begin{align*}(\textbf g_k^{(j)})_{k\in{\mathbb{N}}}=(g^{(j)}_{k,1},\ldots,g^{(j)}_{k,\ell})_{k\in{\mathbb{N}}},\text{ }j\in\{1,\ldots,\ell\},\end{align*} $$in $G^{\ell }$ , there exists an infinite $S\subseteq {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ ,(3.1) $$ \begin{align} {\mathop { \mathcal {R}{\text{}\mathrm{lim}}}_{{\alpha\in S^{(\ell)}}}}\, \mu(A_0\cap T_{ g^{(1)}_{\alpha}}A_1\cap \cdots\cap T_{ g^{(\ell)}_{\alpha}}A_{\ell})=\prod_{j=0}^{\ell}\mu(A_j). \end{align} $$ 
(iii) For any $\epsilon>0$ and any $A_0,\ldots ,A_{\ell }\in \mathcal A$ , the set
$$ \begin{align*}&R_{\epsilon}(A_0,\ldots,A_{\ell})\\&\quad=\bigg\{(g_1,\ldots,g_{\ell})\in G^{\ell}\,\bigg\,\,\bigg\mu(A_0\cap T_{g_1}A_1\cap\cdots \cap T_{g_{\ell}}A_{\ell})\prod_{j=0}^{\ell} \mu( A_j)\bigg<\epsilon\bigg\}\end{align*} $$is $\tilde \Sigma _{\ell }^*$ in $G^{\ell }$ . 
(iv) For any $\epsilon>0$ and any $A_0,A_1\in \mathcal A$ , the set $R_{\epsilon }(A_0,A_1)$ is $\Sigma _{\ell }^*$ in G.
Proof. (i) $\implies $ (ii): Note that since $(T_g)_{g\in G}$ is strongly mixing, for any $t\in \{1,\ldots ,\ell \}$ and any $j\in \{1,\ldots ,\ell \}$ , $(T_{g_{k,t}^{(j)}})_{k\in {\mathbb {N}}}$ has the mixing property and that for any t and any $i\neq j$ , $(T_{(g_{k,t}^{(j)}g_{k,t}^{(i)})})_{k\in {\mathbb {N}}}$ also has the mixing property. Thus (ii) follows from Proposition 2.9.
(ii) $\implies $ (iii): By (ii), we have that, for any $\epsilon>0$ , any $A_0,\ldots ,,A_{\ell }\in \mathcal A$ and any $\ell $ nondegenerated and essentially distinct sequences
in $G^{\ell }$ , there exists an $\alpha \in {\mathbb {N}}^{(\ell )}$ such that
which implies that $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ is $\tilde {\Sigma }_{\ell }^*$ .
(iii) $\implies $ (iv): Let $\epsilon>0$ , let $A_0,A_1\in \mathcal A$ and let $(\textbf g^{(1)}_k)_{k\in {\mathbb {N}}}=(g_{k,1}^{(1)},\ldots ,g_{k,\ell }^{(1)})_{k\in {\mathbb {N}}}$ be a nondegenerated sequence in $G^{\ell }$ . In order to prove that $\mathcal R_{\epsilon }(A_0,A_1)$ is $\Sigma _{\ell }^*$ , it suffices to show that for some $\alpha \in {\mathbb {N}}^{(\ell )}$ , $g_{\alpha }^{(1)}\in \mathcal R_{\epsilon }(A_0,A_1)$ .
Note that, for any sequence $(h^{(1)}_k)_{k\in {\mathbb {N}}}$ in G with $\lim _{k\rightarrow \infty }h_k^{(1)}=\infty $ , one can pick sequences $(h^{(2)}_k)_{k\in {\mathbb {N}}},\ldots,$ $(h^{(\ell )}_k)_{k\in {\mathbb {N}}}$ in G with the property that, for any distinct $i,j\in \{1,\ldots ,\ell \}$ ,
Hence, one can find nondegenerated sequences $(\textbf g_k^{(j)})_{k\in {\mathbb {N}}}$ in $G^{\ell }$ , $j\in \{2,\ldots ,\ell \}$ , such that $(\textbf g^{(1)}_k)_{k\in {\mathbb {N}}},\ldots,$ $(\textbf g^{(\ell )}_k)_{k\in {\mathbb {N}}}$ are essentially distinct. By (iii), there exists an $\alpha \in {\mathbb {N}}^{(\ell )}$ for which
This implies that $g^{(1)}_{\alpha }\in R_{\epsilon }(A_0,A_1)$ .
(iv) $\implies $ (i): We will show that, for any $\xi ,\eta \in L_0^2(\mu )=\{f\in L^2(\mu )\,\,\int _Xf\,\text {d}\mu =0\}$ , $\lim _{g\rightarrow \infty }\langle T_g\xi ,\eta \rangle =0$ . To do this, it suffices to prove that for any sequence $(g_k)_{k\in {\mathbb {N}}}$ in G with $\lim _{k\rightarrow \infty }g_k=\infty $ , there exists an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in ${\mathbb {N}}$ such that, for any $\xi ,\eta \in L^2_0(\mu )$ ,
Let $(g_k)_{k\in {\mathbb {N}}}\subseteq G$ with $\lim _{k\rightarrow \infty }g_k=\infty $ . Let $(\textbf g_k)_{k\in {\mathbb {N}}}=(\underbrace {g_k,\ldots ,g_k}_{\ell \text { times}})_{k\in {\mathbb {N}}}$ (note that $(\textbf g_k)_{k\in {\mathbb {N}}}$ is a nondegenerated sequence in $G^{\ell }$ ). We claim that there exist an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in ${\mathbb {N}}$ and a bounded linear operator $V:L^2_0(\mu )\rightarrow L^2_0(\mu )$ such that, if we set $S=\{k_j\,\,j\in {\mathbb {N}}\}$ , the following assertions hold.

(1) For any $\xi ,\eta \in L_0^2(\mu )$ ,
(3.3) $$ \begin{align} \langle V\xi,\eta\rangle =\lim_{j\rightarrow\infty}\langle T_{g_{k_j}}\xi,\eta\rangle. \end{align} $$ 
(2) For any $A_0,A_1\in \mathcal A$ , there exists a real number $r_{A_0,A_1}$ such that
(3.4) $$ \begin{align} {\mathop { \mathcal {R}{\text{}\mathrm{lim}}}_{{\alpha\in S^{(\ell)}}}}\, \mu(A_0\cap T_{g_{\alpha}}A_1)=r_{A_0,A_1}. \end{align} $$
Let $\mathcal D$ be a countable dense subset of $L_0^2(\mu )$ . By a diagonalization argument, one obtains an increasing sequence $(k^{\prime }_j)_{j\in {\mathbb {N}}}$ for which the limit in (3.3) exists for any $\xi ,\eta \in \mathcal D$ . Diagonalizing once more, we can pick a subsequence $(k_j)_{j\in {\mathbb {N}}}$ of $(k^{\prime }_j)_{j\in {\mathbb {N}}}$ for which (3.4) holds for any $A_0,A_1$ from a countable dense subset of $\mathcal A$ . It follows (by a standard approximation argument) that all the limits appearing in (3.3) and (3.4) exist for any $\xi ,\eta \in L_0^2(\mu )$ and any $A_0,A_1\in \mathcal A$ . Notice that (3.3) holds for a unique linear operator V. Since
we have that V is normbounded.
We claim that $V^{\ell }=0$ . To see this, note that, by (iv), for every $A_0,A_1\in \mathcal A$ , $r_{A_0,A_1}=\mu (A_0)\mu (A_1)$ (otherwise we would be able to find an $\epsilon>0$ for which the set $\mathcal R_{\epsilon }(A_0,A_1)$ is not $\Sigma _{\ell }^*$ ). Since the linear combinations of indicator functions are dense in $L^2(\mu )$ , it follows that, for any $f_1,f_2\in L^2(\mu )$ ,
Observe that, by (3.3), $T_gV=VT_g$ for all $g\in G$ . Thus, all the limits appearing in the expression
exist for any $\xi ,\eta \in L_0^2(\mu )$ . Combining (3.3) and (3.5), we obtain that, for any $\xi ,\eta \in L_0^2(\mu )$ ,
proving our claim.
It follows that in order to prove that (3.2) holds, it is enough to show that $L_0^2(\mu )=\text {Ker}(V^{\ell })\subseteq \text {Ker}(V)$ . To do this, we will first show that V is a normal operator. Indeed, for any $\xi ,\eta \in L^2_0(\mu )$ ,
and, hence,
So, for any $\xi \in L_0^2(\mu )$ ,
Now take $t\in {\mathbb {N}}$ , $\eta \in L^2_0(\mu )$ , and set $\xi =V^{t1}\eta $ . Suppose that $\eta \not \in \text {Ker}(V^t)$ . Then $\xi \not \in \text {Ker}(V)$ and, by (3.6), $\langle V^*V\xi ,\xi \rangle \neq 0$ . Applying (3.6) to $V\xi $ , we obtain $\V^2\xi \^2=\V^*V\xi \^2$ . So, since $\langle V^*V\xi ,\xi \rangle \neq 0$ , $V^{t+1}\eta =V^2\xi \neq 0$ . This proves that, for each $t\in {\mathbb {N}}$ , if $\eta \not \in \text {Ker}(V^t)$ , then $\eta \not \in \text {Ker}(V^{t+1})$ . So, $L_0^2(\mu )=\text {Ker}(V^{\ell })\subseteq \text {Ker}(V)$ and, hence, for any $\xi ,\eta \in L^2_0(\mu )$ ,
4 Some ‘diagonal’ results for strongly mixing systems
In order to give the reader the flavor of the main theme of this section, we start by formulating a slightly enhanced form of Theorem 1.4. (This theorem is a rather special case of the results of ‘diagonal’ nature to be proved in this section.)
Proposition 4.1. Let $(X,\mathcal A,\mu , T)$ be a measurepreserving system and let $a_1,\ldots ,a_{\ell }$ be nonzero distinct integers. Then T is strongly mixing if and only if, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set
is $\Sigma _{\ell }^*$ .
We move now to formulations of more general ‘diagonal’ results.
Let $(G,{\kern1pt}+)$ be a countable abelian group, let $(X,{\kern1pt}\mathcal A,{\kern1pt}\mu ,{\kern1pt} (T_g)_{g\in G})$ be a measurepreserving system, let $\ell \in {\mathbb {N}}$ and let $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be homomorphisms. For any $\epsilon>0$ and any $A_0,\ldots ,A_{\ell }\in \mathcal A$ , define
We first give two equivalent formulations of a general result which deals with finitely generated groups.
Theorem 4.2. Let $(G,+)$ be a finitely generated abelian group, let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system and let the homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be such that, for any $j\in \{1,\ldots ,\ell \}$ , $\ker (\phi _j)$ is finite and, for any $i\neq j$ , $\ker (\phi _j\phi _i)$ is also finite. Then $(T_g)_{g\in G}$ is strongly mixing if and only if, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }}(A_0,\ldots ,A_{\ell })$ is $\Sigma _{\ell }^*$ .
Note that if G is a finitely generated abelian group and $\phi :G\rightarrow G$ is a homomorphism, $\ker (\phi )$ is finite if and only if the index of $\phi (G)$ in G is finite. It follows that Theorem 4.2 can be formulated in the following equivalent form.
Theorem 4.3. Let $(G,+)$ be a finitely generated abelian group, let $(X,\mathcal A,\mu , (T_g)_{g\in G})$ be a measurepreserving system and let the homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be such that, for any $j\in \{1,\ldots ,\ell \}$ , the index of $\phi _j(G)$ in G is finite and, for any $i\neq j$ , the index of $(\phi _j\phi _i)$ in G is also finite. Then $(T_g)_{g\in G}$ is strongly mixing if and only if, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , the set $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }}(A_0,\ldots ,A_{\ell })$ is $\Sigma _{\ell }^*$ .
We now formulate and prove variants of Theorems 4.2 and 4.3 which pertain to mixing actions of general (not necessarily finitely generated) countable abelian groups. Unlike Theorems 4.2 and 4.3, the following two theorems are not equivalent. We will provide the relevant counterexamples at the end of this section.
Theorem 4.4. Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ be a strongly mixing system and let the homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be such that, for any $j\in \{1,\ldots ,\ell \}$ , $\ker (\phi _j)$ is finite and, for any $i\neq j$ , $\ker (\phi _j\phi _i)$ is also finite. For any nondegenerated sequence $(\textbf g_k)_{k\in {\mathbb {N}}}=(g_{k,1},\ldots ,g_{k,\ell })_{k\in {\mathbb {N}}}$ in $G^{\ell }$ , there exists an infinite set $S\subseteq {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell } \in \mathcal A$ ,
Equivalently, for any $A_0,\ldots ,A_{\ell } \in \mathcal A$ and any $\epsilon>0$ , the set $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }}(A_0,\ldots ,A_{\ell })$ is $\Sigma _{\ell }^*$ .
Proof. Since, for any distinct $i,j\in \{1,\ldots ,\ell \}$ , $\ker (\phi _j)$ and $\ker (\phi _j\phi _i)$ are both finite, we have for each $t\in \{1,\ldots ,\ell \}$ ,
For each $j\in \{1,\ldots ,\ell \}$ , let
Then the sequences $(\textbf g^{(1)}_k)_{k\in {\mathbb {N}}},\ldots ,(\textbf g^{(\ell )}_k)_{k\in {\mathbb {N}}}$ are nondegenerated and essentially distinct. By Theorem 3.1(ii), there exists an infinite set $S\subseteq {\mathbb {N}}$ such that, for any $A_0,\ldots ,A_{\ell }\in \mathcal A$ ,
Remark 4.5. The goal of this remark is to indicate an alternative way of proving Theorem 4.4. Let G and $\phi _1,\ldots ,\phi _{\ell }$ be as in the hypothesis of Theorem 4.4. In §5 we will show that if E is a $\tilde {\Sigma }_{\ell }^*$ set in $G^{\ell }$ , then $\{g\in G\,\,(\phi _1(g),\ldots ,\phi _{\ell }(g))\in E\}$ is a $\Sigma _{\ell }^*$ set in G (see Proposition 5.22). Thus, for any measurepreserving system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ , any $A_0,\ldots ,A_{\ell }\in \mathcal A$ and any $\epsilon>0$ , if $R_{\epsilon }(A_0,\ldots ,A_{\ell })$ is a $\tilde {\Sigma }_{\ell }^*$ set, then $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }}(A_0,\ldots ,A_{\ell })$ is a $\Sigma _{\ell }^*$ set. One can now invoke Theorem 3.1(iii).
The next result complements Theorem 4.4. Note that it provides a somewhat stronger version of one of the directions in Theorem 4.3.
Theorem 4.6. Let $(G,+)$ be a countable abelian group, let $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ be a measurepreserving system and let the homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ be such that at least one of $\phi _1(G)$ , $\phi _2(G)$ , $(\phi _2\phi _1)(G)$ has finite index in G. If, for all $A_0,\ldots ,A_{\ell }\in \mathcal A$ and all $\epsilon>0$ , the set $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }}(A_0,\ldots ,A_{\ell })$ is $\Sigma _{\ell }^*$ , then $(T_g)_{g\in G}$ is strongly mixing.
Proof. We will assume that $(\phi _2\phi _1)(G)$ has finite index in G; the other two cases can be handled similarly. For any $A_1,A_2\in \mathcal A$ and any $\epsilon>0$ , we have
By our assumption, for any $\epsilon>0$ and any $A_1,A_2\in \mathcal A$ , the set $R^{\phi _2\phi _1}_{\epsilon }(A_1,A_2)$ is a $\Sigma _{\ell }^*$ set and hence, by Theorem 3.1(iv), $(T_{(\phi _2\phi _1)(g)})_{g\in G}$ is strongly mixing.
We will now prove that $(T_g)_{g\in G}$ is strongly mixing by showing that for any sequence $(g_k)_{k\in {\mathbb {N}}}$ in G with $\lim _{k\rightarrow \infty }g_k=\infty $ , there exists an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in ${\mathbb {N}}$ with the property that for any $A_0,A_1\in \mathcal A$ ,
Let $(g_k)_{k\in {\mathbb {N}}}$ be a sequence in G with $\lim _{k\rightarrow \infty }g_k=\infty $ . By assumption, $(\phi _2\phi _1)(G)$ has finite index in G, so there exist an increasing sequence $(k_j)_{j\in {\mathbb {N}}}$ in ${\mathbb {N}}$ and an element $\tau \in G$ for which $\{g_{k_j}+\tau \,\,j\in {\mathbb {N}}\}\subseteq (\phi _2\phi _1)(G)$ . Since $(T_{(\phi _2\phi _1)(g)})_{g\in G}$ is strongly mixing, for any $A_0,A_1\in \mathcal A$ ,
completing the proof.
The following proposition shows that the assumption made in Theorem 4.2 that G is finitely generated cannot be removed.
Proposition 4.7. Let $G=\bigoplus _{k\in {\mathbb {N}}} {\mathbb {Z}}$ and let $\ell \in {\mathbb {N}}$ . There exist a measurepreserving system $(X,\mathcal A,\mu ,(T_g)_{g\in G})$ and homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ satisfying (a) for any $j\in \{1,\ldots ,\ell \}$ , $\ker (\phi _j)$ is finite, and (b) for any $i\neq j$ , $\ker (\phi _j\phi _i)$ is also finite, and such that every set of the form $R_{\epsilon }^{\phi _1,\ldots ,\phi _{\ell }} (A_0,\ldots ,A_{\ell })$ is $\Sigma _{\ell }^*$ but $(T_g)_{g\in G}$ is not strongly mixing.
Proof. We will only carry out the proof for $\ell =2$ ; the general case can be handled similarly. Let $\phi _1:G\rightarrow G$ be the homomorphism given by
Note that $\phi _1$ is injective (and so $\ker (\phi _1)$ is trivial).
Let $X=\{0,1\}^G$ be endowed with the product topology, let $\mu $ be the $(\tfrac 12,\tfrac 12)$ product measure on $\mathcal A=\text {Borel}(X)$ and, for each $g\in G$ , let $S_g:X\rightarrow X$ be the map defined by $(S_g(x))(h)=x(h+g)$ . The system $(X,\mathcal A,\mu ,(S_g)_{g\in G})$ is strongly mixing. Define a measurepreserving Gaction $(T_g)_{g\in G}$ on $(X,\mathcal A,\mu )$ by
and let $\phi _2:G{\kern1pt}\rightarrow{\kern1pt} G$ be defined by $\phi _2(g){\kern1pt}={\kern1pt}2\phi _1(g)$ . Note that, for any $g{\kern1pt}={\kern1pt}(a_1,a_2,\ldots ){\kern1pt}\in{\kern1pt} G$ ,
So, for any $\epsilon>0$ and any $A_0,A_1,A_2\in \mathcal A$ ,
It follows from Theorem 4.4 that every set of the form
is $\Sigma _2^*$ and hence, by (4.1), for any $A_0,A_1,A_2$ and any $\epsilon>0$ , $R_{\epsilon }^{\phi _1,\phi _2}(A_0,A_1,A_2)$ is $\Sigma _2^*$ .
Noting that for each $k\in {\mathbb {N}}$ , $T_{(k,0,0,\ldots )}=S_{(0,0,\ldots )}$ is the identity map on X, we see that $(T_g)_{g\in G}$ is not strongly mixing. We are done.
The next result shows that Theorem 4.3 cannot be extended to arbitrary countable abelian groups.
Proposition 4.8. Let $G=\bigoplus _{k\in {\mathbb {N}}} {\mathbb {Z}}$ and let $\ell \in {\mathbb {N}}$ . There exist a strongly mixing system $(X,\mathcal A,\mu , (T_g)_{g\in G})$ and homomorphisms $\phi _1,\ldots ,\phi _{\ell }:G\rightarrow G$ satisfying (a) for any $j\in \{1,\ldots ,\ell \}$ ,