2.1 Splitting results

Our first main concern is to resolve the incompatibility between [Reference Donoso, Koutsogiannis and Sun7] and [Reference Ferré Moragues9], and improve the method in [Reference Donoso, Koutsogiannis and Sun7], to obtain an extension of the results in [Reference Ferré Moragues9].

Before we state our first result, we need to introduce some notation.

For $d,L\in \mathbb {N}$ , the polynomial $q=(q_1,\ldots ,q_d): \mathbb {Z}^L\to \mathbb {Z}^d$ is non-constant if some $q_i$ is non-constant. Here we mean that each $q_i$ is a member of $\mathbb {Q}[x_1,\ldots ,x_L]$ with ${q_i(\mathbb {Z}^L)\subseteq \mathbb {Z}.}$ The degree of q is defined as the maximum of the degrees of the $q_i$ terms.

The polynomials $p_{1},\ldots ,p_{k}\colon \mathbb {Z}^{L}\to \mathbb {Z}^{d}$ are called essentially distinct if they are non-constant and $p_{i}-p_{j}$ is non-constant for all $i\neq j$ . (In general, a polynomial ${q\colon \mathbb {Z}^{L}\to \mathbb {Z}^{d}}$ has rational coefficients (that is, vectors with rational coordinates).)

For a subset A of $\mathbb {Q}^{d}$ , we denote The following subgroups of ${\mathbb Z}^{d}$ play an important role in this paper.

Our main result provides an affirmative answer to Question 1.1 under finitely many ergodicity assumptions on the groups $G_{i,j}(\mathbf {p})$ , which generalizes [Reference Ferré Moragues9, Theorem 1.5]. We say that the group $G_{i,j}(\mathbf {p})$ is ergodic for $\mu $ if any function $f \in L^2(\mu )$ that is $T_{a}$ -invariant for all $a \in G_{i,j}(\mathbf {p})$ is constant.

The definition of a D-step nilsequence will be given in §3.1. We say that $a\colon {\mathbb Z}^L \to {\mathbb C}$ is a nullsequence if for any Følner sequence $(I_N)_{N\in {\mathbb N}}, \lim _{N \to \infty } {1}/{|I_N|} \sum _{n \in I_N} |a(n)|^2=0.$

Theorem 2.2. (Decomposition theorem under finitely many ergodicity assumptions)

For $d,k,K,L\in \mathbb {N},$ let $\mathbf {p}=(p_{1},\ldots ,p_{k}),$ where $p_{1},\ldots ,p_{k}\colon \mathbb {Z}^{L}\to \mathbb {Z}^{d}$ is a family of essentially distinct polynomials of degree at most $K,$ and let $(X,\mathcal {B},\mu , (T_{n})_{n\in {\mathbb Z}^{d}})$ be a ${\mathbb Z}^{d}$ -system. If $G_{i,j}(\mathbf {p})$ is ergodic for $\mu $ for all $0\leq i,j\leq k,i\neq j,$ then for all $f_0,\ldots ,f_k \in L^{\infty }(\mu )$ , the multicorrelation sequence

can be decomposed as a sum of a uniform limit of D-step nilsequences and a nullsequence, where $D\in {\mathbb N}$ is a constant depending only on $d,k,K,L.$

We refer the reader to Remark 5.14 for a further discussion on the constant D. Also, note that Theorem 2.2 goes beyond Question 1.1 as it deals with multivariable polynomial iterates (that is, $L>1$ ).

Example 2.3. It was proved in [Reference Ferré Moragues9, Theorem 1.5] that for any probability space $(X,\mathcal {B},\mu )$ and commuting transformations $T_{1},\ldots ,T_{k}$ acting on X, if $T_i$ and $T_iT_j^{-1}$ are ergodic (for all i and all $j \neq i$ , respectively), then for all $f_0,\ldots ,f_k \in L^{\infty }(\mu )$ , the multicorrelation sequence

can be decomposed as a sum of a uniform limit of k-step nilsequences plus a nullsequence. While Theorem 2.2 does not specify the step D of the nilsequence, a quick argument shows that, in this case, one can indeed take $D=k$ (see Remark 6.1 for details).

The following example shows that Theorem 2.2 is stronger than [Reference Ferré Moragues9, Theorem 1.6], which deals with single variable essentially distinct polynomial iterates.

Example 2.4. Let $(X,\mathcal {B},\mu ,T_{1},\ldots ,T_{6})$ be a system with commuting transformations $T_{1},\ldots ,T_{6}$ and $f_{0},f_{1},\ldots ,f_{4}\in L^{\infty }(\mu )$ . Using [Reference Ferré Moragues9, Theorem 1.6], we have that the multicorrelation sequence

(9) $$ \begin{align} \alpha(n)=\int_X f_0\cdot T_{1}^{n^{2}}T_{2}^{n}f_1\cdot T_{1}^{n^{2}}T_{3}^{n}f_2\cdot T_{4}^{n^{3}}f_3\cdot T_{5}^{n^{3}}T_{6}^{n}f_4\,d\mu \end{align} $$

can be decomposed as the sum of a uniform limit of nilsequences and a nullsequence if $T^{a_{1}}_{1}\cdots T^{a_{6}}_{6}$ is ergodic for all $(a_{1},\ldots ,a_{6})\in {\mathbb Z}^{6}\backslash \{\mathbf {{0}}\}$ . In contrast, via Theorem 2.2, one can get the same conclusion by only assuming that $T_{1},T_{2}T_{3}^{-1}, T_{4},T_{5},T_{4}T^{-1}_{5}$ are ergodic. (Indeed, denoting , and $e_{i}$ the vector whose ith entry is 1 and all other entries are 0, since $\mathbf {{p}}=((n^2,n,0,0,0,0),(n^2,0,n,0,0,0)$ , $(0,0,0,n^3,0,0)$ , $(0,0,0,0,n^3,n)),$ we have that $G_{1,0}(\mathbf {p})=G_{2,0}(\mathbf {p})=G(e_{1})$ , $G_{1,3}(\mathbf {p})= G_{2,3}(\mathbf {p})=G_{3,0}(\mathbf {p})\ =\ G(e_{4})$ , $G_{1,4}(\mathbf {p})\ =\ G_{2,4}(\mathbf {p})\ =\ G_{4,0}(\mathbf {p})\ =\ G(e_{5})$ , ${G_{1,2}(\mathbf {p})\ =\ G(e_{2}-e_{3})}$ , $G_{3,4}(\mathbf {p})=G(e_{4}-e_{5})$ .)

2.2 Convergence to the expected limit

In [Reference Donoso, Koutsogiannis and Sun7, Theorem 1.4], the first, third, and fourth authors proved the following case of Conjecture 1.3. If $T_{1},\ldots ,T_{k}$ are commuting transformations acting on a probability space $(X,\mathcal {B},\mu )$ , then $(T_{1}^{p(n)},\ldots ,T_{k}^{p(n)})_{n\in \mathbb {Z}^{L}}$ is jointly ergodic for $\mu $ if and only if $((T_{1}\times \cdots \times T_{k})^{p(n)})_{n\in \mathbb {Z}^{L}}$ is ergodic for $\mu ^{\otimes k}$ and $T_{i}T^{-1}_{j}$ is ergodic for $\mu $ for all $i\neq j$ . In this paper, we further extend this result.

Note that the subconditions in condition (C2) are consistent with those in Conjecture 1.3. However, the reason we provide an alternative set of equivalent subconditions in condition (C2’) is that these subconditions are easier to check in practice.

We now give some examples to illustrate Theorem 2.5. The first one is for polynomials of distinct degrees.

We remark that Example 2.6 can also be proved by using arguments from [Reference Chu, Frantzikinakis and Host6]. We next present two examples in which some polynomials have the same degree and so cannot be recovered by the methods of [Reference Chu, Frantzikinakis and Host6].

Another direction for the joint ergodicity problem is verifying whether condition (C1) implies condition (C2) in Conjecture 1.3. Namely, assume that $(T_{p_{1}(n)}\times \cdots \times T_{p_{k}(n)})_{n\in \mathbb {Z}^{L}}$ is ergodic for $\mu ^{\otimes k}$ to find a condition, say (P), of certain sequences of actions to be ergodic, under which we have that $(T_{p_{1}(n)},\ldots ,T_{p_{k}(n)})_{n\in \mathbb {Z}^{L}}$ is jointly ergodic for $\mu $ . By combining existing results from [Reference Host and Kra18, Reference Johnson20] (see also [Reference Donoso, Koutsogiannis and Sun7, Proposition 1.2]), (P) can be taken to be ‘ $T_{g}$ is ergodic for $\mu $ for all $g\in {\mathbb Z}^{d}\backslash \{\mathbf {{0}}\}$ ’. Denoting $p_{i}(n)=\sum _{v\in {\mathbb N}_0^{L},0\leq \vert v\vert \leq K}b_{i,v}n^{v}$ for some $b_{i,v}\in \mathbb {Q}^{d}$ and $K\in {\mathbb N}_0$ , this result was extended in [Reference Donoso, Koutsogiannis and Sun7, Theorem 1.3], where the previous property is replaced by ‘ $T_{g}$ is ergodic for $\mu $ for all g that belongs to the finite set R’, where

$$ \begin{align*} R=\bigcup_{0< \vert v\vert\leq K}\{b_{i,v}, b_{i,v}-b_{j,v}\colon 1\leq i, j\leq k\}\backslash\{\mathbf{0}\}.\end{align*} $$

In this paper, we replace the latter condition with an even weaker one.

The last example for this section reflects the stronger nature of the previous theorem compared to what was previously known.

2.3 Strategy of the paper

The central ingredient in proving the main results of the paper (Theorems 2.2, 2.5, and 2.9) is to find proper characteristic factors for the limit of the average in equation (4), that is, sub- $\sigma $ -algebras $\mathcal {D}_{1},\ldots ,\mathcal {D}_{k}$ of $\mathcal {B}$ such that the average in equation (4) remains invariant if we replace each $f_{i}$ by its conditional expectation (see below for the definition) with respect to $\mathcal {D}_{i}$ . An important type of characteristic factor, called the Host–Kra characteristic factor, was invented in [Reference Host and Kra18] to study multiple averages for ${\mathbb Z}$ -systems (see below for the definition of these factors). This concept was generalized to systems with commuting transformations in [Reference Host17] (see also [Reference Sun31]).

To introduce the main tool used in our results (Theorem 2.11), special cases of which have been studied extensively in the past (see for example [Reference Chu, Frantzikinakis and Host6, Reference Frantzikinakis and Kra14, Reference Host17, Reference Host and Kra18, Reference Johnson20]), we need to introduce the machinery of Host–Kra seminorms and factors.

Host–Kra seminorms and their associated factors are arguably the main tools used to analyze the behavior of multiple averages and correlation sequences. In what follows, we give general results about these seminorms and factors, following the notation used in [Reference Donoso, Koutsogiannis and Sun7].

We first recall the notions of a factor and of the conditional expectation with respect to a factor. We say that the ${\mathbb Z}^d$ -system $(Y,\mathcal {D},\nu ,(S_{g})_{g\in {\mathbb Z}^d})$ is a factor of $(X,\mathcal {B},\mu ,(T_{g})_{g\in {\mathbb Z}^d})$ if there exists a measurable map $\pi \colon X\to Y$ such that $\mu (\pi ^{-1}(A))=\nu (A)$ for all $A\in \mathcal {D}$ , and $\pi \circ T_{g}=S_{g}\circ \pi $ for all $g\in {\mathbb Z}^{d}$ .

A factor $(Y,\mathcal {D},\nu ,(S_{g})_{g\in {\mathbb Z}^d})$ of $(X,\mathcal {B},\mu ,(T_{g})_{g\in {\mathbb Z}^d})$ can be identified with an invariant sub- $\sigma $ -algebra $\mathcal {B}'$ of $\mathcal {B}$ by setting . Given two $\sigma $ -algebras, $\mathcal {B}_1$ and $\mathcal {B}_2$ , their joining $\mathcal {B}_1\vee \mathcal {B}_2$ is the $\sigma $ -algebra generated by $B_1\cap B_2$ for all $B_1\in \mathcal {B}_1$ and $B_2\in \mathcal {B}_2$ , that is, the smallest $\sigma $ -algebra containing both $\mathcal {B}_1$ and $\mathcal {B}_2$ .

Given a factor $\pi \colon (X,\mathcal {B},\mu ) \to (Y,\mathcal {D},\nu )$ and a function $f\in L^{2}(\mu )$ , the conditional expectation of f with respect to Y is the function $g\in L^{2}(\nu ),$ which we denote by $\mathbb {E}(f \mid Y),$ with the property

$$ \begin{align*} \int_A g\circ \pi \, d\mu=\int_A f \ d\mu\quad \textrm{for all } A \in \pi^{-1}(\mathcal{D}).\end{align*} $$

Let $(X,\mathcal {B},\mu )$ be a probability space and let $\mathcal {B}_1$ be a sub- $\sigma $ -algebra of $\mathcal {B}$ . The relatively independent joining of $(X,\mathcal {B},\mu )$ with itself with respect to $\mathcal {B}_1$ is the probability space $(X\times X, \mathcal {B} \otimes \mathcal {B}, \mu \times _{\mathcal {B}_1} \mu )$ , where the measure $\mu \times _{\mathcal {B}_1} \mu $ is given by the formula:

$$ \begin{align*} \int_{X \times X} f_1 \otimes f_2 \, d(\mu \times_{\mathcal{B}_1} \mu)=\int_X \mathbb{E}(f_1|\mathcal{B}_1) \mathbb{E}(f_2| \mathcal{B}_1) \, d\mu, \end{align*} $$

for all $f_1, f_{2} \in L^{\infty }(\mu )$ .

For a G-system $\mathbf {X}=(X,\mathcal {B},\mu ,(T_{g})_{g\in G})$ , if H is a subgroup of $G,$ we denote by $\mathcal {I}(H)(\mathbf {X})$ the set of $A\in \mathcal {B}$ such that $T_{g}A=A$ for all $g\in H$ . When there is no confusion, we write $\mathcal {I}(H)$ .

For a ${\mathbb Z}^d$ -system $(X,\mathcal {B},\mu ,(T_{g})_{g\in {\mathbb Z}^d})$ and $H_1,\ldots ,H_k$ subgroups of ${\mathbb Z}^d,$ define

$$ \begin{align*}\mu_{H_{1}}=\mu\times_{\mathcal{I}(H_{1})}\mu\end{align*} $$

and for $k>1$ , let

$$ \begin{align*}\mu_{H_{1},\ldots,H_{k}}=\mu_{H_{1},\ldots,H_{k-1}}\times_{\mathcal{I}(H_{k}^{[k-1]})}\mu_{H_{1},\ldots,H_{k-1}},\end{align*} $$

where $H^{[k-1]}_{k}$ denotes the subgroup of $({\mathbb Z}^{d})^{2^{k-1}}$ consisting of all the elements of the form $(h_{k},\ldots , h_{k})$ ( $2^{k-1}$ copies of $h_{k}$ ) for some $h_{k}\in H_{k}$ . The characteristic factor $Z_{H_{1},\ldots ,H_{k}}(\mathbf {X})$ is defined to be the sub- $\sigma $ -algebra of $\mathcal {B}$ characterized by

for all $f \in L^{\infty }(\mu )$ , where $X^{[k]}=X\times \cdots \times X$ ( $2^k$ copies of X), $\vert \epsilon \vert =\epsilon _{1}+\cdots +\epsilon _{k}$ for $\epsilon =(\epsilon _{1},\ldots ,\epsilon _{k})\in \{0,1\}^{k}$ , and $\mathcal {C}^{2r+1}f=\overline {f},$ the complex conjugate of f, $\mathcal {C}^{2r}f=f$ for all $r\in {\mathbb Z}$ . The quantity $\lvert \hspace{-1pt}\lvert \hspace{-1pt}\lvert f\rvert \hspace{-1pt}\rvert \hspace{-1pt}\rvert _{H_1,\ldots ,H_k}$ denotes the Host–Kra seminorm of f with respect to the subgroups $H_1,\ldots ,H_k$ . Similar to the proof of [Reference Host17, Lemma 4] or [Reference Host and Kra18, Lemma 4.3], one can show that $Z_{H_{1},\ldots ,H_{k}}(\mathbf {X})$ is well defined.

Theorem 2.11. Let $d,k,K,L\in \mathbb {N}, \mathbf {p}=(p_{1},\ldots , p_{k}), p_{1},\ldots , p_{k}\in \mathbb {Z}^{L}\to \mathbb {Z}^{d}$ be a family of essentially distinct polynomials of degrees at most $K.$ There exists $D\in \mathbb {N}_0$ depending only on $d,k,K,L$ such that for every $\mathbb {Z}^{d}$ -system $\mathbf {X}=(X,\mathcal {B},\mu , (T_{n})_{n\in \mathbb {Z}^{d}})$ , every $f_{1},\ldots , f_{k}\in L^{\infty }(\mu )$ , and every Følner sequence $(I_{N})_{N\in {\mathbb N}}$ of ${\mathbb Z}^{L}$ , if $f_{i}$ is orthogonal to the Host–Kra characteristic factor ${Z}_{\{G_{i,j}(\mathbf {p})\}^{\times D}_{0\leq j\leq k, j\neq i}}(\mathbf {X})$ for some $1\leq i\leq k$ (that is, the conditional expectation of $f_{i}$ under ${Z}_{\{G_{i,j}(\mathbf {p})\}^{\times D}_{0\leq j\leq k, j\neq i}}(\mathbf {X})$ is $0$ ), then we have that

(10) $$ \begin{align} \lim_{N\to\infty}\frac{1}{|I_N|}\sum_{n\in I_{N}}\prod_{i=1}^k T_{p_{i}(n)}f_i=0. \end{align} $$

In particular, if for some $1\leq i\leq k$ , $G_{i,j}(\mathbf {p})$ is ergodic for $\mu $ for all $0\leq j\leq k, j\neq i$ and $f_{i}$ is orthogonal to the Host–Kra characteristic factor ${Z}_{({\mathbb Z}^{d})^{\times kD}}(\mathbf {X})$ , then equation (10) holds.

It is worth noting that the factor ${Z}_{\{G_{i,j}(\mathbf {p})\}^{\times D}_{0\leq j\leq k, j\neq i}}(\mathbf {X})$ we obtain in Theorem 2.11 is not optimal, but it is good enough for our purposes.

A special case of Theorem 2.11 was proved in [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1]. In particular, Theorem 2.11 generalizes [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1] in the following ways.

1. (I) The characteristic factor obtained in Theorem 2.11 is of finite step, whereas that in [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1] is of infinite step.

2. (II) The groups $G_{i,j}(\mathbf {p})$ involved in Theorem 2.11 are larger than those in [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1], which makes the characteristic factors in Theorem 2.11 smaller.

We remark that the aforementioned technical distinctions have significant influences on the applications of Theorem 2.11. First, the essential reason why one cannot directly use [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1] to improve [Reference Ferré Moragues9, Theorem 1.5] is that the method used in [Reference Ferré Moragues9] requires a characteristic factor of finite step. This problem is resolved by generalization (I), enabling us to extend [Reference Ferré Moragues9, Theorem 1.5] in this paper. Second, [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1] does not provide a strong enough characteristic factor in certain circumstances. For example, in the case of Example 2.6, [Reference Chu, Frantzikinakis and Host6, Theorem 6.5] suggests that the Host–Kra seminorms controlling equation (10) depend only on the transformations $T_{1},\ldots ,T_{k}$ , whereas the upper bound provided by [Reference Donoso, Koutsogiannis and Sun7, Theorem 5.1] depends not only on the transformations $T_{1},\ldots ,T_{k}$ but also on many compositions of them. With the help of generalizations (I) and (II), we are able to obtain (and generalize) the aforementioned upper bound of [Reference Chu, Frantzikinakis and Host6, Theorem 6.5].

Roughly speaking, the achievement of generalization (I) relies on a sophisticated development of a Bessel-type inequality first obtained by Tao and Ziegler in [Reference Tao and Ziegler33, Proposition 3.6]. The most technical part of this paper is the approach we use to get generalization (II). In [Reference Donoso, Koutsogiannis and Sun7], a method was introduced to keep track of the coefficients of the polynomials while running a variation of the polynomial exhaustion technique (PET) induction. However, the tracking provided there is not strong enough to imply Theorem 2.11. To overcome this difficulty, we introduce more sophisticated machinery to have a better control of the coefficients.

The paper is organized as follows. We provide some background material in §3. In §4, we present the variation of PET induction that we use. In §5, we address how generalizations (I) and (II) above can be achieved with Propositions 5.2 and 5.4, which improve Propositions 5.6 and 5.5 of [Reference Donoso, Koutsogiannis and Sun7], respectively. We conclude the section by proving Theorem 2.11. This is the bulk of the paper. In §6, we use Theorem 2.11 to deduce Theorems 2.2, 2.5, and 2.9, which are the main results of the paper. We conclude with some discussions on future directions in §7.

2.4 Notation

We denote by $\mathbb {N}, \mathbb {N}_0, \mathbb {Z}, \mathbb {Q}, \mathbb {R}$ , and $\mathbb {C}$ the sets of positive integers, non-negative integers, integers, rational numbers, real numbers, and complex numbers, respectively. If X is a set and $d\in \mathbb {N}$ , $X^d$ denotes the Cartesian product $X\times \cdots \times X$ of d copies of X.

We will denote by $e_i$ the vector that has $1$ as its ith coordinate and $0$ elsewhere. We use in general lower-case letters to symbolize both numbers and vectors but bold letters to symbolize vectors of vectors to highlight this exact fact. The only exception to this convention is the vector $\mathbf {0}$ (that is, the vector with coordinates only $0$ ) which we always symbolize in bold.

Throughout this article, we use the following notation for averages. Let $(a(n))_{n\in {\mathbb Z}^L}$ be a sequence of complex numbers, or a sequence of measurable functions on a probability space $(X,\mathcal {B},\mu )$ . We let:

(we use the symbol $\square $ to highlight the fact that the averages are taken along the boxes $[-N,N]^{L}$ );

and

It is worth noticing that if the limit $\lim _{N\to \infty } \mathbb {E}_{n\in I_N} a(n)$ exists for all Følner sequences (in ${\mathbb Z}^L$ ), then this limit does not depend on the chosen Følner sequence.

We also consider iterated averages. Let $(a(h_{1},\ldots ,h_{s}))_{h_{1},\ldots ,h_{s}\in {\mathbb Z}^L}$ be a multiparameter sequence. We let

and adopt similar conventions for $\mathbb {E}_{h_{1},\ldots ,h_{s}\in \mathbb {Z}^{L}}$ , ${\overline {\mathbb {E}}}^{\square }_{h_{1},\ldots ,h_{s}\in \mathbb {Z}^{L}}$ , and $\mathbb {E}^{\square }_{h_{1},\ldots ,h_{s}\in \mathbb {Z}^{L}}$ .

We end this section by recalling the notion of a system indexed by a countable abelian group $(G,+)$ . We say that a tuple $(X,\mathcal {B},\mu ,(T_{g})_{g\in G})$ is a G-measure-preserving system (or a G-system) if $(X,\mathcal {B},\mu )$ is a probability space and $T_{g}\colon X\to X$ are measurable, measure-preserving transformations on X such that $T_{e_{G}}=\mathrm {id}$ ( $e_G$ is the identity element of G) and $T_{g}\circ T_{h}=T_{g+h}$ for all $g,h\in G$ . A G-system will be called ergodic if for any $A\in \mathcal {B}$ such that $T_{g}A=A$ for all $g\in G$ , we have that $\mu (A)\in \{0,1\}$ . In this paper, we are mostly concerned about ${\mathbb Z}^{d}$ -systems and $L^2(\mu )$ -norm limits of (multiple) ergodic averages. For the corresponding norm, when it is clear from the context, we will write $\Vert {\cdot} \Vert _2$ instead of $\Vert {\cdot} \Vert _{L^2(\mu )}$ .

3.1 Nilsystems, nilsequences, and structure theorem

Let $X=N/\Gamma $ , where N is a (k-step) nilpotent Lie group and $\Gamma $ is a discrete cocompact subgroup of N. Let $\mathcal {B}$ be the Borel $\sigma $ -algebra of $X, \mu $ the normalized Haar measure on $X,$ and for $n\in {\mathbb Z}^{d},$ let ${T_{n}\colon X\to X}$ with $T_{n}x=b_{n}\cdot x$ for some group homomorphism $n\mapsto b_{n}$ from ${\mathbb Z}^{d}$ to N. We say that $\mathbf {X}=(X,\mathcal {B},\mu ,(T_{n})_{n\in {\mathbb Z}^{d}})$ is a (k-step) ${\mathbb Z}^{d}$ -nilsystem. For $k\geq 1$ , we say that $(a_{n})_{n\in {\mathbb Z}^{d}} \subseteq \mathbb {C}$ is a (k-step) ${\mathbb Z}^{d}$ -nilsequence if there exist a (k-step) ${\mathbb Z}^{d}$ -nilsystem $(X,\mathcal {B},\mu ,(T_{n})_{n\in {\mathbb Z}^{d}})$ , a function $F \in C(X)$ and $x\in X$ such that $a_{n}=F(T_{n}x)$ for all $n\in {\mathbb Z}^{d}$ . For $k=0$ , a $0$ -step nilsequence is a constant sequence. An important reason which makes the Host–Kra characteristic factors powerful is their connection with nilsystems. The following is a slight generalization of [Reference Ziegler36, Theorem 3.7] (see [Reference Griesmer16, Lemma 4.4.3 and Theorem 4.10.1], or Proposition 3.1(ii) and [Reference Sun31, Theorem 3.7]), which is a higher dimensional version of the Host–Kra structure theorem [Reference Host and Kra18].

3.2 Bessel’s inequality

An essential difference in the study of multiple ergodic averages between ${\mathbb Z}$ -systems and ${\mathbb Z}^{d}$ -systems is that in the former case, one can usually bound the average by some Host–Kra seminorm of a function f appearing in the average, whereas in the latter, one can only bound the averages by an average of a family of Host–Kra seminorms of f. To overcome this difficulty, inspired by the work of Tao and Ziegler [Reference Tao and Ziegler33], in this subsection, we derive an upper bound for expressions of the form ${\overline {\mathbb {E}}}_{i\in I}\lvert \hspace{-1pt}\lvert \hspace{-1pt}\lvert f\rvert \hspace{-1pt}\rvert \hspace{-1pt}\rvert _{H_{i,1},\ldots ,H_{i,s}}$ , where I is a finite set and $H_{i,j}$ are subgroups of ${\mathbb Z}^{d}$ .

The proof of the following statement is similar to [Reference Tao and Ziegler33, Corollary 1.22].

Proposition 3.6. (Bessel’s inequality)

Let $t\in {\mathbb N}$ , $(X,\mathcal {B},\mu ,(T_{g})_{g\in \mathbb {Z}^{d}})$ be a $\mathbb {Z}^{d}$ -system, I be a finite set of indices, and $H_{i,j}, i\in I, 1\leq j\leq t$ be subgroups of ${\mathbb Z}^{d}$ . Then for all $f\in L^{\infty }(\mu )$ ,

$$ \begin{align*}\mathbb{E}_{i\in I}\| \mathbb{E}(f\vert Z_{H_{i,1},\ldots,H_{i,t}})\|^{2}_{2} \leq \Vert f\Vert_{2}\cdot(\mathbb{E}_{i,j\in I}\Vert \mathbb{E}(f\vert Z_{\{H_{i,i'}+H_{j,j'}\}_{1\leq i',j'\leq t}})\Vert^{2}_{2})^{1/2}.\end{align*} $$

Proof. For convenience, let . Then,

$$ \begin{align*}\mathbb{E}_{i\in I}\Vert \mathbb{E}(f\vert Z_{H_{i,1},\ldots,H_{i,t}})\Vert^{2}_{2}=\langle f,\mathbb{E}_{i\in I}f_{i}\rangle\end{align*} $$

which, by the Cauchy–Schwarz inequality, is bounded by

$$ \begin{align*}\Vert f\Vert_{2}\cdot\vert\mathbb{E}_{i,j\in I}\langle f_{i},f_{j}\rangle\vert^{1/2}. \end{align*} $$

By [Reference Tao and Ziegler33, Corollary 1.21], $L^{\infty }(Z_{H_{i,1},\ldots ,H_{i,t}})$ and $L^{\infty }(Z_{H_{j,1},\ldots ,H_{j,t}})$ are orthogonal on the orthogonal complement of $L^{\infty }(Z_{\{H_{i,i'}+H_{j,j'}\}_{1\leq i',j'\leq t}})$ , and hence

$$ \begin{align*}\langle f_{i},f_{j}\rangle=\| \mathbb{E}(f\vert Z_{\{H_{i,i'}+H_{j,j'}\}_{1\leq i',j'\leq t}})\|_{2}^{2},\end{align*} $$

and we have the conclusion.

By repeatedly using Proposition 3.6, we have the following inequality.

Corollary 3.7. Let $t,s\in {\mathbb N}$ , $(X,\mathcal {B},\mu ,(T_{g})_{g\in \mathbb {Z}^{d}})$ be a $\mathbb {Z}^{d}$ -system, I be a finite set of indices, and $H_{i,j}, i\in I, 1\leq j\leq t,$ be subgroups of ${\mathbb Z}^{d}$ . Then for all $f\in L^{\infty }(\mu )$ , we have

$$ \begin{align*}(\mathbb{E}_{i\in I}\Vert \mathbb{E}(f\vert Z_{H_{i,1},\ldots,H_{i,t}})\Vert^{2}_{2})^{ 2^s} \!\leq \Vert f\Vert^{2\cdot 2^s-2}_{2}\cdot \mathbb{E}_{i_{1},\ldots,i_{2^s}\in I}\big\Vert \mathbb{E}\big(f\big| Z_{\{\sum_{j=1}^{2^s}H_{i_{j},i^{\prime}_{j}}\}_{1\leq i^{\prime}_{1},\ldots,i^{\prime}_{2^s}\leq t}}\big)\big\Vert^{2}_{2}.\end{align*} $$

The next proposition provides an upper bound for $\mathbb {E}_{i\in I}\lvert \hspace{-1pt}\lvert \hspace{-1pt}\lvert f\rvert \hspace{-1pt}\rvert \hspace{-1pt}\rvert _{H_{i,1},\ldots ,H_{i,t}}$ which can be combined with the previous two statements.

Proposition 3.8. Let $t\in {\mathbb N}$ , $(X,\mathcal {B},\mu ,(T_{g})_{g\in \mathbb {Z}^{d}})$ be a $\mathbb {Z}^{d}$ -system, I be a finite set of indices, and $H_{i,j}, i\in I, 1\leq j\leq t$ be subgroups of ${\mathbb Z}^{d}$ . Then, for all $f\in L^{\infty }(\mu )$ , with $\Vert f\Vert _{L^{\infty }(\mu )}\leq 1$ ,

$$ \begin{align*}\mathbb{E}_{i\in I}\lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}}\leq (\mathbb{E}_{i\in I}\Vert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\Vert^{2}_{2})^{1/2^{t}}.\end{align*} $$

Proof. Note that

(11) $$ \begin{align} \lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}}\leq \Vert f\Vert_{L^{2^{t}}(\mu)}\leq \Vert f\Vert_{2}^{1/2^{t-1}}. \end{align} $$

Also, for all i, we have

$$ \begin{align*} \lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}}&\leq \lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f-\mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}} + \lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}}\\ &= \lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}}, \end{align*} $$

so

(12) $$ \begin{align} &\mathbb{E}_{i\in I}\lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}} \leq \mathbb{E}_{i\in I}\lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}}) \rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{H_{i,1},\ldots,H_{i,t}} \nonumber\\ &\quad\leq \mathbb{E}_{i\in I}\Vert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\Vert_{2}^{1/2^{t-1}} \leq (\mathbb{E}_{i\in I}\Vert \mathbb{E}(f|Z_{H_{i,1},\ldots,H_{i,t}})\Vert^{2}_{2})^{1/2^{t}}, \end{align} $$

as was to be shown.

3.3 General properties of subgroups of ${\mathbb Z}^d$ and properties of polynomials

Recall that for a subset A of $\mathbb {Q}^{d}$ , we denote $G(A):= \text {span}_{{\mathbb Q}} \{a\in A\}\cap {\mathbb Z}^{d}.$ Next, we summarize some properties of these sets.

In the remainder of the section, we provide some algebraic lemmas that will be used later in the paper. For a set $E\subseteq {\mathbb Z}^d,$ we define its upper Banach density (or just upper density when there is no confusion) by If the limit exists, we say that its value is the Banach density (or just density) of E. The proof of the following lemma is routine (see also [Reference Donoso, Koutsogiannis and Sun7, Lemma 2.11] for a more general version).

Lemma 3.12. Let $v_{i}\in {\mathbb Z}^{L}, 1\leq i\leq k$ and U be a subset of ${\mathbb Z}^{k}$ of positive density. Then,

(13) $$ \begin{align} G\bigg(\bigg\{\sum_{1\leq i\leq k}h_{i} v_{i}\colon \mathbf{h}=(h_1,\ldots,h_k)\in U\bigg\}\bigg)=G(\{v_{i}\colon 1\leq i\leq k\}).\end{align} $$

Proof. Note that in equation (13), the right-hand side clearly includes the left-hand side. To prove the converse inclusion, it suffices to show that

(14) $$ \begin{align} \text{span}_{\mathbb{Q}}\{\mathbf{h}\colon \mathbf{h}\in U\}=\mathbb{Q}^{k}. \end{align} $$

Since U has positive density, it cannot be contained in any hyperplane of ${\mathbb Q}^k$ , so it must have at least k elements that are linearly independent over ${\mathbb Q}$ . Thus, equation (14) follows immediately.

4.1 The van der Corput lemma

The standard tool used in reducing the complexity of polynomial families of iterates is van der Corput’s lemma (also known as ‘van der Corput’s trick’). We will use the following variation of it, the proof of which can be found in [Reference Donoso, Koutsogiannis and Sun7, Lemma 2.2].

Lemma 4.1. (van der Corput lemma)

Let $(a(n;h_1,\ldots ,h_s))_{(n;h_1,\ldots ,h_s)\in ({\mathbb Z}^{L})^{s+1}}, s\in \mathbb {N}_0,$ be a sequence bounded by $1$ in a Hilbert space $\mathcal {H}$ . Then, for all $\tau \in \mathbb {N}_0$ ,

$$ \begin{align*} &\overline{\mathbb{E}}^{\square}_{h_1,\ldots,h_s\in {\mathbb Z}^{L}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \| \mathbb{E}_{n\in I_{N}} a(n;h_1,\ldots,h_s) \|^{2\tau}\\ &\,\leq\! 4^{\tau}\overline{\mathbb{E}}^{\square}_{h_1,\ldots,h_s,h_{s+1}\in {\mathbb Z}^{L}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\!\varlimsup_{N\to \infty} \kern-1pt\vert \mathbb{E}_{n\in I_{N}}\! \langle a(n+h_{s+1};h_1,\ldots,h_s) , a(n;h_1,\ldots,h_s) \rangle \vert^{\tau}\kern-1pt. \end{align*} $$

We also provide two applications of Lemma 4.1 for later use. The first one is to get an upper bound for single averages with polynomial iterates and a polynomial exponential weight. Let and recall Definition 3.13 for the polynomial $\Delta ^{K}P$ .

Lemma 4.3. Let $P\colon {\mathbb Z}^{L}\to {\mathbb R}$ and $p\colon {\mathbb Z}^{L}\to {\mathbb Z}^{d}$ be polynomials. For all $K\in {\mathbb N}_0$ and $\tau \in {\mathbb N}$ , there exists $C_{K,\tau }>0$ such that for every $\mathbb {Z}^{d}$ -system, $(X,\mathcal {B},\mu , (T_{g})_{g\in \mathbb {Z}^{d}}),$ and ${f\in L^{\infty }(\mu )}$ bounded by 1, we have

$$ \begin{align*} &{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\Vert\mathbb{E}_{n\in I_{N}}\exp(P(n))T_{p(n)}f\Vert_{2}^{2\tau} \\&\quad\leq C_{K,\tau}\overline{\mathbb{E}}^{\square}_{\mathbf{h}=(h_{1},\ldots,h_{K})\in({\mathbb Z}^{L})^{K}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\Vert\mathbb{E}_{n\in I_{N}}\exp(\Delta^{K}P(n,\mathbf{h}))T_{\Delta^{K}p(n,\mathbf{h})}f\Vert^{\tau}_{2}. \end{align*} $$

Proof. When $K=0$ , there is nothing to prove. We now assume that the relation holds for some $K\in {\mathbb N}_0$ and we show it for $K+1$ . Using Lemma 4.1 and the T-invariance of $\mu $ , we get

$$ \begin{align*} &\overline{\mathbb{E}}^{\square}_{\mathbf{h}=(h_1,\ldots,h_{K})\in ({\mathbb Z}^{L})^{K}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \| \mathbb{E}_{n\in I_{N}} \exp(\Delta^{K}P(n,\mathbf{h}))T_{\Delta^{K}p(n,\mathbf{h})}f \|^{2\tau}_{2}\\ &\;\kern-1pt\leq\! 4^{\kern-0.7pt\tau}\overline{\mathbb{E}}^{\square}_{\mathbf{h}=(h_1,\ldots,h_{K+1})\in ({\mathbb Z}^{L})^{K+1}}\kern-2pt{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\!\kern-2pt\varlimsup_{N\to \infty}\! \vert \mathbb{E}_{n\in I_{N}} \kern-2pt\!\int_{\kern-0.5pt X}\!\kern-1pt\exp(\Delta^{\kern-1.2pt K+1}\kern-1pt P(n,\mathbf{h})) T_{\kern-0.5pt \Delta^{\kern-0.5pt K+1}p(n,\mathbf{h})}f\!\cdot\! \overline{f}\,d\mu \vert^{\kern-1pt\tau} \\&\;\kern-1pt\leq 4^{\tau}\overline{\mathbb{E}}^{\square}_{\mathbf{h}=(h_1,\ldots,h_{K+1})\in ({\mathbb Z}^{L})^{K+1}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\!\varlimsup_{N\to \infty}\!\Vert \mathbb{E}_{n\in I_{N}} \exp(\Delta^{K+1}P(n,\mathbf{h}))T_{\Delta^{K+1}p(n,\mathbf{h})}f\Vert^{\tau}_{2}, \end{align*} $$

and hence the result (the constant that appears depends only on $\tau $ and K).

The second application of Lemma 4.1 provides an upper bound for single averages, with linear iterates and an exponential weight evaluated at a linear polynomial, on a product system. The proof is inspired by [Reference Donoso, Koutsogiannis and Sun7, Lemma 5.2] and [Reference Host and Kra19, Proposition 2.9].

Lemma 4.4. Let $(X,\mathcal {B},\mu )$ be a probability space, $k,L\in {\mathbb N}$ and $T_{i,j}, 1\leq i\leq k, 1\leq j\leq L$ be commuting measure-preserving transformations on X. Denote $S_{j}=T_{1,j}\times \cdots \times T_{k,j}$ for $1\leq j\leq L$ . Let $G_{i}$ be the group generated by $T_{i,1},\ldots ,T_{i,L}$ . Then, for any polynomial $P\colon {\mathbb Z}^{L}\to {\mathbb R}$ of degree 1 and $f_{1},\ldots ,f_{k}\in L^{\infty }(\mu )$ bounded by 1, we have that

(15) $$ \begin{align} {\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\Vert\mathbb{E}_{n\in I_{N}}\exp(P(n))R_n f\Vert_{L^{2}(\mu^{\otimes k})}\leq 2\min_{1\leq i\leq k}\lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f_{i}\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert_{G_{i}^{\times 2}}, \end{align} $$

where $f=f_{1}\otimes \cdots \otimes f_{k}$ and for

Proof. Fix $1\leq i\leq k$ and let $P(n)=a\cdot n+b$ for some $a\in {\mathbb R}^{L},b\in {\mathbb R}$ . Then, by Lemma 4.1 for $\tau =2$ and $s=0$ , the fourth power of the left-hand side of equation (15) is bounded by

$$ \begin{align*} & 16\cdot\mathbb{E}^{\square}_{h\in{\mathbb Z}^{L}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\bigg\vert\int_{X^{k}}\mathbb{E}_{n\in I_{N}}\exp(P(n+h)-P(n))R_{n+h}f\cdot R_{n}\overline{f}\,d\mu^{\otimes k}\bigg\vert^{2} \\&\quad = 16\cdot\mathbb{E}^{\square}_{h\in{\mathbb Z}^{L}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\bigg\vert\int_{X^{k}}\mathbb{E}_{n\in I_{N}}\exp(a\cdot h)R_{h}f\cdot \overline{f}\,d\mu^{\otimes k}\bigg\vert^{2} \\&\quad = 16\cdot\mathbb{E}^{\square}_{h\in{\mathbb Z}^{L}}\bigg\vert\int_{X^{k}}R_{h}f\cdot \overline{f}\,d\mu^{\otimes k}\bigg\vert^{2} \\&\quad = 16\cdot\mathbb{E}^{\square}_{h=(h_{1},\ldots,h_{L})\in{\mathbb Z}^{L}}\bigg\vert\int_{X^{k}}\bigotimes_{i=1}^{k}\bigg((\prod_{j=1}^{L}T_{i,j}^{h_{j}})f_{i}\cdot \overline{f}_{i}\bigg)\,d\mu^{\otimes k}\bigg\vert^{2} \\&\quad \leq 16\cdot\mathbb{E}^{\square}_{h=(h_{1},\ldots,h_{L})\in{\mathbb Z}^{L}}\bigg\vert\int_{X}\bigg(\prod_{j=1}^{L}T_{i,j}^{h_{j}}\bigg) f_{i}\cdot \overline{f}_{i}\,d\mu\bigg\vert^{2} \\&\quad \leq 16\cdot\mathbb{E}^{\square}_{h=(h_{1},\ldots,h_{L})\in{\mathbb Z}^{L}}\bigg\vert\int_{X}\mathbb{E} \bigg((\prod_{j=1}^{L}T_{i,j}^{h_{j}})f_{i}\cdot \overline{f}_{i}\vert \mathcal{I}(G_{i})\bigg)\,d\mu\bigg\vert^{2} \\ &\quad =16\lvert\hspace{-1pt}\lvert\hspace{-1pt}\lvert f_{i}\rvert\hspace{-1pt}\rvert\hspace{-1pt}\rvert^{4}_{G_{i}^{\times 2}}, \end{align*} $$

from where the result follows.

4.2 The van der Corput operation

To review the PET induction scheme, we will follow, and slightly modify, the approach from [Reference Donoso, Koutsogiannis and Sun7]. To this end, we extend the definitions that we have already given on the polynomial families of interest (see the beginning of §2.1), taking into account that we treat the first L-tuple of variables of the polynomials differently.

Before we list the steps of the van der Corput operation, we will present the manipulations of the inner product in Lemma 4.1 in a simple example where we have three essentially distinct polynomial iterates $(p_1(n),p_2(n),p_3(n))=(n^2,2n,n),$ to show how, by repeatedly running the van der Corput trick, we get an expression of linear iterates. This will be extended to general expressions in Theorem 4.9. Here, we want to study, for bounded by $1$ functions $f_1, f_2, f_3,$ the average of the sequence $a(n)=T_1^{n^2}f_1\cdot T_2^{2n}f_2\cdot T_2^n f_3.$ Notice that we can write this sequence as a $\mathbb {Z}^2$ -action, $a(n)=T_{(n^2,0)}f_1\cdot T_{(0,2n)}f_2\cdot T_{(0,n)}f_3$ for the triple of polynomials $((n^2,0),(0,2n),(0,n)).$ Using Lemma 4.1, we have

$$ \begin{align*} &{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \| \mathbb{E}_{n\in I_{N}} a(n) \|^{2} \\ &\quad \leq 4\overline{\mathbb{E}}^{\square}_{h_1\in {\mathbb Z}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \vert \mathbb{E}_{n\in I_{N}} \langle a(n+h_{1}) , a(n) \rangle \vert\\ &\quad = 4\overline{\mathbb{E}}^{\square}_{h_1\in {\mathbb Z}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \vert \mathbb{E}_{n\in I_{N}}\int_X T_1^{(n+h_1)^2}f_1\cdot T_2^{2(n+h_1)}f_2\cdot T_2^{n+h_1}f_3\cdot T_1^{n^2}\bar{f}_1\\ &\qquad\times T_2^{2n}\bar{f}_2 \cdot T_2^{n}\bar{f}_3 \,d\mu\vert. \end{align*} $$

Using the fact that $T_2$ is measure-preserving, we compose by $T_2^{-n}$ (notice that n is the polynomial of the minimum degree in the expression) to get

$$ \begin{align*} & 4\overline{\mathbb{E}}^{\square}_{h_1\in {\mathbb Z}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \Bigg \vert \mathbb{E}_{n\in I_{N}}\int_X \bar{f}_3\cdot T_2^{h_1}f_3\cdot T_1^{n^2+2h_1 n}T_2^{-n}(T_1^{h_1^2}f_1)\\ &\quad \times T_1^{n^2}T_2^{-n}\bar{f}_1\cdot T_2^{n}(T_2^{2h_1}f_2\cdot \bar{f}_2)\;d\mu\Bigg\vert, \end{align*} $$

where we grouped the functions with the same linear terms.

Using the Cauchy–Schwarz inequality (to discard the terms that have iterates independent of n), the previous relation is bounded by

$$ \begin{align*}\overline{\mathbb{E}}^{\square}_{h_1\in {\mathbb Z}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty} \| \mathbb{E}_{n\in I_{N}}T_1^{n^2+2h_1 n}T_2^{-n}(T_1^{h_1^2}f_1)\cdot T_1^{n^2}T_2^{-n}\bar{f}_1\cdot T_2^{n}(T_2^{2h_1}f_2\cdot \bar{f}_2)\|.\end{align*} $$

Exactly because of the grouping of the terms of the same linear iterates, the resulting polynomial iterates, $(n^2+2h_1 n,-n), (n^2,-n), (0,n),$ have the property that they are non-constant and that their pairwise differences are non-constant (this will lead us below to the notion of the ‘essentially distinct’ vector-valued polynomials).

Similarly, skipping the details, using Lemma 4.1, composing with $T_2^{-n}$ (the polynomial $(0,n)$ is of minimum ‘degree’—see below for the definition of the degree of a vector-valued polynomial), the square of the previous quantity can be bounded by

$$ \begin{align*} & \overline{\mathbb{E}}^{\square}_{(h_{1},h_2)\in {\mathbb Z}^{2}}\!{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\!\kern-1pt\varlimsup_{N\to \infty}\!\| \mathbb{E}_{n\in I_{N}}\!T_1^{n^2+2(h_1+h_2)n}T_2^{-2n}(T_1^{(h_1+h_2)^2}\!T_2^{-h_2}f_1)\! \cdot T_1^{n^2+2h_1n} T_2^{-2n}(T_1^{h_1^2}\bar{f}_1) \\ & \quad\times T_1^{n^2+2h_2n}T_2^{-2n}(T_1^{h_2^2}T_2^{-h_2}\bar{f}_1) \cdot T_1^{n^2}T_2^{-2n}f_1 \|. \end{align*} $$

Note that the iterates in the previous relation are ‘essentially distinct’ for ‘almost all’ tuples $(h_{1},h_{2})\in \mathbb {Z}^{2}$ .

Analogously, using Lemma 4.1 once more, noticing that all the resulting terms in the expression will have the factor $T_1^{n^2}T_2^{-2n},$ where $(n^2,-2n)$ is the polynomial of minimum ‘degree’, we can bound, composing with the term $T_1^{-n^2}T_2^{2n}$ , the square of the previous relation by

$$ \begin{align*} & \overline{\mathbb{E}}^{\square}_{(h_{1},h_2,h_{3})\in {\mathbb Z}^{3}}{\sup_{\substack{ (I_{N})_{N\in\mathbb{N}} \\ \mathrm{ F\unicode{xf8} lner\, seq.} }}}\varlimsup_{N\to \infty}\| \mathbb{E}_{n\in I_{N}} T_1^{2(h_1+h_2+h_3)n}(T_1^{(h_1+h_2+h_3)^2}T_2^{-h_2-2h_3}f_1)\\ &\quad \times T_1^{2(h_2+h_3)n}(T_1^{(h_2+h_3)^2}T_2^{-h_2-2h_3}\bar{f}_1)\cdot T_1^{2(h_1+h_3)n}(T_1^{(h_1+h_3)^2}T_2^{-2h_3}\bar{f}_1) \cdot T_1^{2h_3 n}(T_1^{h_3^2}T_2^{-2h_3}f_1) \\ &\quad \times T_1^{2(h_1+h_2)n}(T_1^{(h_1+h_2)^2}T_2^{-h_2}\bar{f}_1) \cdot T_1^{2h_2 n}(T_1^{h_2^2}T_2^{-h_2}f_1) \cdot T_1^{2h_1 n}(T_1^{h_1^2}f_1)\;d\mu\|. \end{align*} $$

The iterates in this last relation are linear with distinct coefficients for ‘almost all’ tuples $(h_{1},h_{2},h_{3})\in \mathbb {Z}^{3}$ . So, the eighth power of the initial expression is bounded by the previous relation.

The previous example leads naturally to the following notions.

Definition 4.5. For a polynomial $p(n;h_{1},\ldots ,h_{s})\colon (\mathbb {Z}^{L})^{s+1}\to \mathbb {Z}$ , we denote by $\deg (p)$ the degree of p with respect to n (for example, for $s=1, L=2$ , the degree of $p(n_{1},n_{2};h_{1,1},h_{1,2})=h_{1,1}h_{1,2}n_{1}^{2}+h_{1,1}^{5}n_{2}$ is 2).

For a polynomial $p(n;h_{1},\ldots ,h_{s})=(p_{1}(n;h_{1},\ldots ,h_{s}),\ldots ,p_{d}(n;h_{1},\ldots ,h_{s}))\colon (\mathbb {Z}^{L})^{s+1}\to \mathbb {Z}^{d},$ we let $\deg (p)=\max _{1\leq i\leq d}\deg (p_{i})$ and we say that p is non-constant if $\deg (p)>0$ (that is, some $p_i$ is a non-constant function of n), otherwise, we say that p is constant. The polynomials $q_{1},\ldots ,q_{k}\colon (\mathbb {Z}^{L})^{s+1}\to \mathbb {Z}^{d}$ are called essentially distinct if they are non-constant and $q_i-q_j$ is non-constant for all $i\neq j$ . Finally, for a tuple $\mathbf {q}=(q_{1},\ldots ,q_{k}),$ we let $\deg (\mathbf {q})=\max _{1\leq i\leq k}\deg (q_{i}).$ (For clarity, we use non-bold letters for vectors (of polynomials) and bold letters for vectors of vectors (of polynomials).)

Let $(X,\mathcal {B},\mu ,(T_{g})_{g\in \mathbb {Z}^{d}})$ be a $\mathbb {Z}^{d}$ -system, $q_{1},\ldots ,q_{k}\colon (\mathbb {Z}^{L})^{s+1}\to \mathbb {Z}^{d}$ be polynomials, and $g_{1},\ldots , g_{k}\colon X\times (\mathbb {Z}^{L})^{s}\to \mathbb {R}$ be functions such that $g_{m}(\cdot ;h_{1},\ldots ,h_{s})$ is an $L^{\infty }(\mu )$ function bounded by $1$ for all $h_{1},\ldots ,h_{s}\in \mathbb {Z}^L, 1\leq m\leq k$ . If $\mathbf {q}=(q_{1},\ldots ,q_{k})$ and $\mathbf {g}=(g_{1},\ldots ,g_{k}),$ we say that $A=(L,s,k,\mathbf {g},\mathbf {q})$ is a PET-tuple, and for $\tau \in {\mathbb N}_0$ , we set

(16)

We define $\deg (A)=\deg (\mathbf {q})$ , and say that A is non-degenerate if $\mathbf {q}$ is a family of essentially distinct polynomials (for convenience, $\mathbf {q}$ will be called non-degenerate as well). For $1\leq m\leq k$ , the tuple A is m-standard for $f\in L^\infty (\mu )$ if $\deg (A)=\deg (q_{m})$ and $g_{m}(x;h_{1},\ldots ,h_{s})=f(x)$ for every $x,h_1,\ldots ,h_s$ . That is, f is the mth function in $\mathbf {g}$ , only depending on the first variable, and the polynomial $q_m$ that acts on f is of the highest degree. (Here, we say m-standard for f to highlight the function of interest as, after running the vdC-operation, the position of the functions in the expression we deal with changes.) The tuple A will be called semi-standard for f if there exists $1\leq m\leq k$ such that $g_{m}(x;h_{1},\ldots ,h_{s})=f(x)$ for every $x,h_1,\ldots ,h_s$ . In this case, we do not require the function f to have a specific position in $\mathbf {g}$ nor that the polynomial acting on f be of the highest degree.

As an example, for a $\mathbb {Z}$ -system $(X,\mathcal {B},\mu ,T),$ take $L=s=1, k=3, q_1(n,h)=n^3, q_2(n,h)=3n^2h, q_3(n,h)=3nh^2,$ and, for $f,g\in L^\infty (\mu ),$ let $g_1(x,h)=f(x), g_2(x,h)=g(x),$ and $g_3(x,h)=T^{h^3}f(x).$

Then, we have that A is 1-standard for $f,$ semi-standard for f and $g,$ and, for $\kappa \in \mathbb {N}_0$ ,

For each non-degenerate PET-tuple $A{\kern-1pt}={\kern-1.5pt}(L,{\kern-0.5pt}s,{\kern-0.5pt}k,{\kern-0.5pt}\mathbf {g},{\kern-0.5pt}\mathbf {q})$ and polynomial $q\colon{\kern-1.2pt} (\mathbb {Z}^{L})^{s+1}{\kern-1.2pt}\to{\kern-1.2pt} \mathbb {Z}^{d}$ , we define the vdC-operation, $\partial _{q}A$ , according to the following three steps. (Actually, the vdC-operation can be defined for any PET-tuple, not just for non-degenerate ones. Similarly, being a procedure that reduces complexity, PET induction can be applied to any family of polynomials. As the expressions of interest in this paper correspond to non-degenerate tuples, we consider only this case.)

Step 1: For all $1\leq m\leq k$ , let $g^{\ast }_{m}=g^{\ast }_{m+k}=g_{m},$ and $q^{\ast }_1,\ldots ,q^{\ast }_{2k} \colon (\mathbb {Z}^{L})^{s+2}\to \mathbb {Z}^{d}$ be the polynomials defined as

$$ \begin{align*}\displaystyle q^{\ast}_m(n;h_1,\ldots,h_{s+1})=\left\{\!\begin{array}{ll} q_m(n+h_{s+1};h_1,\ldots,h_{s})-q(n;h_1,\ldots,h_{s}), & 1\leq m\leq k,\\ q_{m-k}(n;h_1,\ldots,h_{s})\!-\!q(n;h_1,\ldots,h_{s}), & k\!+\!1\!\leq\! m\!\leq\! 2k,\end{array} \right. \end{align*} $$

that is, we subtract the polynomial q from the first k polynomials after we have shifted by $h_{s+1}$ the first L variables, and for the second k ones, we subtract q. (In practice, this q will be one of the $q_i$ terms of minimum degree.) Denote $\mathbf {q}^{\ast }=(q^{\ast }_{1},\ldots ,q^{\ast }_{2k})$ .

Step 2: We remove from $q^{\ast }_{1},\ldots ,q^{\ast }_{2k}$ the polynomials which are constant and the associated functions $g_i^\ast $ in the expression (we group all these terms together and we see the resulting term as a single constant one, in terms of n). As we already saw in the example at the beginning of this subsection, this is justified via the use of the Cauchy–Schwarz inequality and the fact that the functions $g_m$ are bounded. Then we put the non-constant ones in groups $J_{i}=\{\tilde {q}_{i,1},\ldots ,\tilde {q}_{i,t_{i}}\}, 1\leq i\leq k'$ for some $k', t_{i}\in \mathbb {N}$ such that any two polynomials are essentially distinct if and only if they belong to different groups. Next, we write $\tilde {q}_{i,j}(n;h_{1},\ldots ,h_{s+1})=\tilde {q}_{i,1}(n;h_{1},\ldots ,h_{s+1})+\tilde {p}_{i,j}(h_{1},\ldots ,h_{s+1})$ for some polynomial $\tilde {p}_{i,j}$ for all $1\leq j\leq t_{i}, 1\leq i\leq k'$ . For convenience, we also relabel what remains, as some of the initial terms may have been removed because of the grouping of the polynomials, of the $g^{\ast }_{1},\ldots , g^{\ast }_{2k}$ accordingly as $\tilde {g}_{i,j}$ for all $1\leq j\leq t_{i}, 1\leq i\leq k'$ .

Step 3: For all $1\leq i\leq k'$ , let $q^{\prime}_{i}=\tilde {q}_{i,1}$ and

$$ \begin{align*}g^{\prime}_{i}(x;h_{1},\ldots,h_{s+1})=\tilde{g}_{i,1}(x;h_{1},\ldots,h_{s+1})\prod^{t_{i}}_{j=2}T_{\tilde{p}_{i,j}(h_{1},\ldots,h_{s+1})}\tilde{g}_{i,j}(x;h_{1},\ldots,h_{s+1}).\end{align*} $$

We set $\mathbf {q}'=(q^{\prime}_{1},\ldots ,q^{\prime}_{k'})$ , $\mathbf {g}'=(g^{\prime}_{1},\ldots ,g^{\prime}_{k'})$ and we denote the new PET-tuple by . (Here we abuse the notation by writing $\partial _{q}A$ to denote any such tuple, obtained from Steps 1 to 3. Strictly speaking, $\partial _{q}A$ is not uniquely defined as the order of the grouping of $q^{\prime}_{1},\ldots ,q^{\prime}_{2k}$ in Step 2 is ambiguous. However, this is done without loss of generality, since the order does not affect the value of $S(\partial _{q}A, \cdot )$ .) It follows from the construction that $\partial _{q}A$ is non-degenerate because each $q_i'$ comes from a distinct grouping $J_i.$

If $q=q_{t}$ for some $1\leq t\leq k$ , we write $\partial _{t}\mathbf {q}$ instead of $\mathbf {q}'$ to highlight the fact that we have subtracted the polynomial $q_t$ ; we also write $\partial _{t}A$ instead of $\partial _{q_{t}}A$ to lighten the notation.

Example 4.7. Let $\mathbf {p}=(p_{1},p_{2})$ with $p_{1}, p_{2}\colon {\mathbb Z}\to {\mathbb Z}^{d}$ the polynomials given by $p_{i}(n)=b_{i,2}n^{2}+b_{i,1}n$ for some $b_{i,1},b_{i,2}\in {\mathbb Z}^{d}$ for $1\leq i\leq 2$ with $b_{1,2},b_{2,2},b_{1,2}- b_{2,2}\neq \mathbf {0}$ (hence, we have that $L=1, s=0,$ and $k=2$ ). We will calculate $\partial _2\mathbf {{p}}.$ Subtracting $p_2$ in the Step 1 of the vdC operation, we have that $\partial _2\mathbf {{p}}=(q_{1},q_{2},q_{3})$ is a tuple of three polynomials, $q_{1},q_{2},q_{3}\colon {\mathbb Z}^{2}\to {\mathbb Z}^{d},$ given by

$$ \begin{align*}\nonumber q_{1}(n,h_1)&=p_1(n+h_1)-p_2(n)\\ &=(b_{1,2}-b_{2,2})n^{2}+2b_{1,2}nh_1+(b_{1,1}-b_{2,1})n+b_{1,1}h_1 +b_{1,2}h_1^{2},\\ q_{2}(n,h_1)&=p_2(n+h_1)-p_2(n)=2b_{2,2}nh_1+b_{2,1}h_1+b_{2,2}h_1^{2}, \\ q_{3}(n,h_1)&=p_1(n)-p_2(n)=(b_{1,2}-b_{2,2})n^{2}+(b_{1,1}-b_{2,1})n, \end{align*} $$

where we removed one essentially constant polynomial in Step 2 of the vdC operation. (Here we use $q_i$ terms instead of $p^{\prime}_i$ terms in the first step to ease the notation of Example 5.8 that is given in the next section.) Notice that here we have $L=1, s'=1,$ and $k'=3.$

Actually, as in the example at the beginning of this subsection, after using a series of vdC operations, one can convert $\mathbf {{p}}$ into a PET-tuple of linear polynomials. Indeed, if we run the vdC operation once more by subtracting $q_2$ in Step 1 of the vdC operation, we have that $\partial _2\partial _2\mathbf {{p}}=(q^{\prime}_{1},\ldots ,q^{\prime}_{4})$ is a tuple of four polynomials, $q^{\prime}_{1},\ldots ,q^{\prime}_{4}\colon {\mathbb Z}^{3}\to {\mathbb Z}^{d},$ given by

$$ \begin{align*} q^{\prime}_{1}(n,h_{1},h_{2}) &=(b_{1,2}-b_{2,2})n^{2}+2(b_{1,2}-b_{2,2})nh_{1}+2(b_{1,2}-b_{2,2})nh_{2}\\ &\quad+(b_{1,1}-b_{2,1}) n+r^{\prime}_{1}(h_{1},h_{2}), \\ q^{\prime}_{2}(n,h_{1},h_{2}) &=(b_{1,2}-b_{2,2})n^{2}-2b_{2,2}nh_{1}+2(b_{1,2}-b_{2,2})nh_{2} \\ &\quad +(b_{1,1}-b_{2,1})n +r^{\prime}_{2}(h_{1},h_{2}), \\ q^{\prime}_{3}(n,h_{1},h_{2})&=(b_{1,2}-b_{2,2})n^{2}+2(b_{1,2}-b_{2,2})nh_{1}+(b_{1,1}-b_{2,1})n+r^{\prime}_{3}(h_{1},h_{2}), \\ q^{\prime}_{4}(n,h_{1},h_{2})&=(b_{1,2}-b_{2,2})n^{2}-2b_{2,2}nh_{1}+(b_{1,1}-b_{2,1})n+r^{\prime}_{4}(h_{1},h_{2}), \end{align*} $$

where $r^{\prime}_{i}\colon {\mathbb Z}^{2}\to {\mathbb Z}^{d}, 1\leq i\leq 4,$ are polynomials in $h_1, h_2$ , and we removed two essentially constant polynomials (that is, $q_{2}(n,h_1)-q_{2}(n,h_1)$ and $q_{2}(n+h_{2},h_1)-q_{2}(n,h_1)$ ) in Step 2 of the vdC operation.

Finally, if we apply the vdC operation again by subtracting $q^{\prime}_4$ in Step 1 of the vdC operation, we have that $\partial _4\partial _2\partial _2\mathbf {{p}}=(q^{\prime \prime }_{1},\ldots ,q^{\prime \prime }_{7})$ is a tuple of seven polynomials, $ q^{\prime \prime }_{1},\ldots ,q^{\prime \prime }_{7}\colon {\mathbb Z}^{4}\to {\mathbb Z}^{d},$ given by

$$ \begin{align*} q^{\prime\prime}_{1}(n,h_{1},h_{2},h_{3}) &=2b_{1,2}nh_{1}\kern1.2pt{+}\kern1.2pt 2(b_{1,2}\kern1.3pt{-}\kern1.3pt b_{2,2})nh_{2}\kern1.3pt{+}\kern1.3pt 2(b_{1,2}\kern1.3pt{-}\kern1.3pt b_{2,2})nh_{3} +r^{\prime\prime}_{1}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{2}(n,h_{1},h_{2},h_{3})&=2(b_{1,2}-b_{2,2})nh_{2}+2(b_{1,2}-b_{2,2})nh_{3} +r^{\prime\prime}_{2}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{3}(n,h_{1},h_{2},h_{3})&=2b_{1,2}nh_{1}+2(b_{1,2}-b_{2,2})nh_{3}+r^{\prime\prime}_{3}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{4}(n,h_{1},h_{2},h_{3})&=2(b_{1,2}-b_{2,2})nh_{3}+r^{\prime\prime}_{4}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{5}(n,h_{1},h_{2},h_{3})&=2b_{1,2}nh_{1}+2(b_{1,2}-b_{2,2})nh_{2}+r^{\prime\prime}_{5}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{6}(n,h_{1},h_{2},h_{3})&=2(b_{1,2}-b_{2,2})nh_{2}+r^{\prime\prime}_{6}(h_{1},h_{2},h_{3}), \\ q^{\prime\prime}_{7}(n,h_{1},h_{2},h_{3})&=2b_{1,2}nh_{1}+r^{\prime\prime}_{7}(h_{1},h_{2},h_{3}), \end{align*} $$

where $r^{\prime \prime }_{i}\colon {\mathbb Z}^{3}\to {\mathbb Z}^{d}, 1\leq i\leq 7,$ are polynomials in $h_1, h_2, h_3$ , and we removed one essentially constant polynomial in Step 2 of the vdC operation. It is clear that $\deg (\partial _4\partial _2\partial _2\mathbf {{p}})=1$ .

The vdC operation provides us with a non-degenerate tuple, the value $S(\cdot , \cdot )$ of which satisfies the following.

The following crucial result (cf. [Reference Donoso, Koutsogiannis and Sun7, Theorem 4.2]) shows that when we start with a PET-tuple which is 1-standard for a function, then, after finitely many vdC operations, we arrive at a new PET-tuple of degree 1 which is still 1-standard for the same function, so we can then use some Host–Kra seminorm to bound the $\limsup $ of the average of interest.

If A is an m-standard PET-tuple for the function $f,$ then, by rearranging the terms if necessary, one can get a new tuple $A'$ which is 1-standard for f with $S(A,\tau )=S(A',\tau )$ . However, if A is semi-standard but not standard for f, then the PET-induction does not work well enough to provide an upper bound for $S(A,\tau )$ in terms of some Host–Kra seminorm of f. To overcome this difficulty one follows [Reference Donoso, Koutsogiannis and Sun7]. More specifically, using [Reference Donoso, Koutsogiannis and Sun7, Proposition 6.3], which is a ‘dimension-increment’ argument, A can be transformed into a new PET-tuple which is 1-standard for f (at the cost of increasing the dimension from L to $2L$ which is harmless for our approach). So, following this procedure, for any fixed function $f,$ we may assume without loss of generality that the corresponding polynomial iterate p is of maximum degree, making the PET-tuple, after potential rearrangement of its terms, $1$ -standard for f. A combination of the previous results will allow us to obtain the required upper bound for each function.