2.1 Basic notation

We start with explaining basic notation used throughout the paper.

The letters ${\mathbb C}, {\mathbb R}, {\mathbb Z}, {\mathbb N}, and {\mathbb N}_0$ stand for the set of complex numbers, real numbers, integers, positive integers, and non-negative integers. With ${\mathbb T}$ , we denote the one-dimensional torus, and we often identify it with ${\mathbb R}/{\mathbb Z}$ or with $[0,1)$ . We let $[N]:=\{1, \ldots , N\}$ for any $N\in {\mathbb N}$ . With ${\mathbb Z}[n]$ , we denote the collection of polynomials with integer coefficients.

For an element $t\in {\mathbb R}$ , we let $e(t):=e^{2\pi i t}$ .

If $a\colon {\mathbb N}^s\to {\mathbb C}$ is a bounded sequence for some $s\in {\mathbb N}$ and A is a non-empty finite subset of ${\mathbb N}^s$ , we let

$$ \begin{align*}\mathop{\mathbb{E}}\limits_{n\in A}\,a(n):=\frac{1}{|A|}\sum_{n\in A}\, a(n). \end{align*} $$

We commonly use the letter $\ell $ to denote the number of transformations in our system or the number of functions in an average while the letter s usually stands for the degree of ergodic seminorms. We normally write tuples of length $\ell $ in bold, e.g. ${\textbf {b}}\in {\mathbb Z}^{\ell }$ , and we underline tuples of length s (or $s+1$ , or $s-1$ ) that are typically used for averaging, e.g. ${\underline {h}}\in {\mathbb Z}^s$ . For a vector ${\textbf {b}}=(b_1,\ldots , b_{\ell })\in {\mathbb Z}^{\ell }$ and a system $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ , we let

$$ \begin{align*} T^{{\textbf{b}}}:=T_1^{b_1}\cdots T_{\ell}^{b_{\ell}}, \end{align*} $$

and we denote the $\sigma $ -algebra of $T^{\textbf {b}}$ invariant functions by ${\mathcal I}(T^{\textbf {b}})$ . For $j\in [\ell ]$ , we set ${\mathbf {e}}_j$ to be the unit vector in ${\mathbb Z}^{\ell }$ in the jth direction, and we let ${\mathbf {e}}_0 = \mathbf {0}$ , so that $T^{{\mathbf {e}}_j} = T_j$ for $j\in [\ell ]$ and $T^{{\mathbf {e}}_0}$ is the identity transformation.

We often write ${\underline {\epsilon }}\in \{0,1\}^s$ for a vector of 0s and 1s of length s. For ${\underline {\epsilon }}\in \{0,1\}^s$ and ${\underline {h}}, {\underline {h}}'\in {\mathbb Z}^s$ , we set:

• • ${\underline {\epsilon }}\cdot {\underline {h}}:=\epsilon _1 h_1+\cdots + \epsilon _s h_s$ ;

• • $\mathopen {}| {\underline {h}}\mathclose {}| := |h_1|+\cdots +|h_s|$ ;

• • ${\underline {h}}^{\underline {\epsilon }} := (h_1^{\epsilon _1}, \ldots , h_s^{\epsilon _s})$ , where $h_j^0:=h_j$ and $h_j^1:=h_j'$ for $j=1,\ldots , s$ .

We let ${\mathcal C} z := \overline {z}$ be the complex conjugate of $z\in {\mathbb C}$ .

For a tuple $\eta \in {\mathbb N}_0^{\ell }$ and $I\subset [\ell ]$ , we define the restriction $\eta |_I := (\eta _i)_{i\in I}$ .

2.2 Ergodic seminorms

We review some basic facts about two families of ergodic seminorms: the Gowers–Host–Kra seminorms and the box seminorms.

2.2.1 Gowers–Host–Kra seminorms

Given a system $(X, {\mathcal X}, \mu ,T)$ , we will use the family of ergodic seminorms $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| \cdot |\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T}$ , also known as Gowers–Host–Kra seminorms, which were originally introduced in [Reference Host and Kra13] for ergodic systems. A detailed exposition of their basic properties can be found in [Reference Host and Kra14, Ch. 8]. These seminorms are inductively defined for $f\in L^{\infty }(\mu )$ as follows (for convenience, we also define $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| \cdot |\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _0$ , which is not a seminorm):

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{0,T}:=\int f\, d\mu, \end{align*} $$

and for $s\in {\mathbb N}_0$ , we let

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s+1,T}^{2^{s+1}}:=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{h\in [H]} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{T; h}f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s,T}^{2^{s}}, \end{align*} $$

where

$$ \begin{align*} \Delta_{T;h}f:=f\cdot T^h\overline{f}, \quad h\in {\mathbb Z}, \end{align*} $$

is the multiplicative derivative of f with respect to T. The limit can be shown to exist by successive applications of the mean ergodic theorem, and for $f\in L^{\infty }(\mu )$ and $s\in {\mathbb N}_0$ , we have $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s,T}\leq \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s+1,T}$ (see [Reference Host and Kra13] or [Reference Host and Kra14, Ch. 8]). It follows immediately from the definition that

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{1,T}=\Vert {\mathbb{E}}(f|{\mathcal I}(T))\Vert_{L^2(\mu)}, \end{align*} $$

where ${\mathcal I}(T):=\{f\in L^2(\mu )\colon Tf=f\}$ . We also have

(9) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s,T}^{2^s}=\lim_{H_1\to\infty}\cdots \lim_{H_s\to\infty}\mathop{\mathbb{E}}\limits_{h_1\in [H_1]}\cdots \mathop{\mathbb{E}}\limits_{h_s\in [H_s]} \int \Delta_{s, T; {\underline{h}}}f\, d\mu, \end{align} $$

where for ${\underline {h}}=(h_1,\ldots , h_s)\in {\mathbb Z}^s$ , we let

$$ \begin{align*} \Delta_{s,T;{\underline{h}}}f:=\Delta_{T;h_1}\cdots \Delta_{T;h_s}f=\prod_{{\underline{\epsilon}}\in \{0,1\}^s}\mathcal{C}^{|{\underline{\epsilon}}|}T^{{\underline{\epsilon}}\cdot {\underline{h}}}f \end{align*} $$

be the multiplicative derivative of f of degree s with respect to T.

It can be shown that we can take any $s'\leq s$ of the iterative limits to be simultaneous limits (that is average over $[H]^{s'}$ and let $H\to \infty $ ) without changing the value of the limit in equation (9). This was originally proved in [Reference Host and Kra13] using the main structural result of [Reference Host and Kra13]; a more ‘elementary’ proof can be deduced from [Reference Bergelson and Leibman4, Lemma 1.12] once the convergence of the uniform Cesàro averages is known (and yet another proof can be found in [Reference Host12, Lemma 1]). For $s':=s$ , this gives the identity

(10) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s,T}^{2^s}=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in [H]^s} \int \Delta_{s, T; {\underline{h}}}f\, d\mu. \end{align} $$

Moreover, for $1\leq s'\leq s$ , we have

(11) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s,T}^{2^{s}}=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in [H]^{s-s'}} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{s-s',T;{\underline{h}}}f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s'}^{2^{s'}}. \end{align} $$

It has been established in [Reference Host and Kra13] for ergodic systems and in [Reference Host and Kra14, Ch. 8, Theorem 14] for general systems that the seminorms are intimately connected with a certain family of factors of the system. Specifically, for every $s\in {\mathbb N}$ , there exists a factor ${\mathcal Z}_s(T)\subseteq {\mathcal X}$ , known as the Host–Kra factor of degree s, with the property that

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s, T} = 0 \quad\text{if and only if } f \text{ is orthogonal to } {\mathcal Z}_{s-1}(T). \end{align*} $$

Equivalently, $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| \cdot |\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T}$ defines a norm on the space $L^2({\mathcal Z}_{s-1}(T))$ (for a proof, see [Reference Host and Kra14, Theorem 15, Ch. 9]).

2.2.2 Box seminorms

More generally, we use analogs of equation (10) defined with regards to several commuting transformations. These seminorms originally appeared in the work of Host [Reference Host12]; their finitary versions are often called box seminorms, and we sometimes employ this terminology. Let $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ be a system. For each $f\in L^{\infty }(\mu )$ , $h\in {\mathbb Z}$ , and ${\textbf {b}} \in {\mathbb Z}^{\ell }$ , we define

$$ \begin{align*} \Delta_{{\textbf{b}}; h} f := f \cdot T^{{\textbf{b}} h} \overline{f} \end{align*} $$

and for ${\underline {h}}\in {\mathbb Z}^s$ and ${\textbf {b}}_1,\ldots , {\textbf {b}}_s\in {\mathbb Z}^{\ell }$ , we let

$$ \begin{align*} \Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}} f := \Delta_{{{\textbf{b}}_1; h_1}}\cdots\Delta_{{{\textbf{b}}_s; h_s}} f = \prod_{{\underline{\epsilon}}\in\{0,1\}^s} {\mathcal C}^{|{\underline{\epsilon}}|} T^{{\textbf{b}}_1 \epsilon_1 h_1 + \cdots + {\textbf{b}}_s \epsilon_s h_s}f. \end{align*} $$

We let

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{\emptyset}:=\int f\, d\mu \end{align*} $$

and

(12) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s+1}}}^{2^{s+1}}:=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{h\in[H]}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{{{\textbf{b}}_{s+1}}; h}f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}}^{2^s}. \end{align} $$

In particular, if ${\textbf {b}}_1 = \cdots = {\textbf {b}}_s:={\textbf {b}}$ , then $ \Delta _{{\textbf {b}}_1, \ldots , {\textbf {b}}_s; {\underline {h}}}=\Delta _{s, T^{\textbf {b}}; {\underline {h}}}$ and $ \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| \cdot |\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_s}=\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| \cdot |\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T^{\textbf {b}}}$ . We remark that these seminorms were defined in a slightly different way in [Reference Host12] and the above identities were established in [Reference Host12, §2.3].

Iterating equation (12), we get the identity

(13) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}}^{2^{s+1}}=\lim_{H_1\to\infty}\cdots \lim_{H_s\to\infty}\mathop{\mathbb{E}}\limits_{h_1\in [H_1]}\cdots \mathop{\mathbb{E}}\limits_{h_s\in [H_s]} \int\Delta_{{{\textbf{b}}_1; h_1}}\cdots\Delta_{{{\textbf{b}}_s; h_s}} f\, d\mu, \end{align} $$

which extends equation (9). In a complete analogy with the remarks made for the Gowers–Host–Kra seminorms, we have the following: using [Reference Host12, Lemma 1] (which implies the convergence of the uniform Cesàro averages over ${\underline {h}}\in {\mathbb Z}^s$ of $\int \Delta _{{{\textbf {b}}_1}, \ldots , {{\textbf {b}}_{s}}; {\underline {h}}} f\, d\mu $ ) and [Reference Bergelson and Leibman4, Lemma 1.12], we get that we can take any $s'\leq s$ of the iterative limits to be simultaneous limits (that is, average over $[H]^{s'}$ and let $H\to \infty $ ) without changing the value of the limit in equation (13). Taking $s'=s$ gives the identity

(14) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}}^{2^s}=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in [H]^s}\,\int \Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}} f\, d\mu. \end{align} $$

More generally, for any $1\leq s'\leq s$ and $f\in L^{\infty }(\mu )$ , we get the identity

(15) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}^{2^s} = \lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in[H]^{s-s'}}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{{\textbf{b}}_{s'+1}, \ldots, {\textbf{b}}_s; {\underline{h}}}f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_{s'}}^{2^{s'}}, \end{align} $$

which generalizes equation (11).

As a consequence of the identity in equation (14), we observe that for any permutation $\sigma :[s]\to [s]$ , we have

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s} = \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_{\sigma(1)}, \ldots, {\textbf{b}}_{\sigma(s)}}, \end{align*} $$

that is, the order of taking the vectors ${\textbf {b}}_1, \ldots , {\textbf {b}}_s$ does not matter.

As an example of a box seminorm that is not a Gowers–Host–Kra seminorm, consider $s=2$ and the vectors ${\mathbf {e}}_1=(1,0)$ , ${\mathbf {e}}_2=(0,1)$ , in which case

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\mathbf{e}}_1},{{\mathbf{e}}_2}}^4 = \lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{h_1, h_2\in [H]^2}\int f\cdot T_1^{h_1}\overline{f}\cdot T_2^{h_2}\overline{f}\cdot T_1^{h_1}T_2^{h_2}f\, d\mu. \end{align*} $$

More generally, for $s=2$ and ${\mathbf {a}}=(a_1,a_2)$ , ${\textbf {b}}=(b_1,b_2)$ , we have

$$ \begin{align*} &\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\mathbf{a}}},{{\textbf{b}}}}^4\\&\quad=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{h_1, h_2\in [H]^2}\int f\cdot T_1^{a_1h_1}T_2^{a_2h_1}\overline{f}\cdot T_1^{b_1h_2}T_2^{b_2h_2}\overline{f}\cdot T_1^{a_1h_1+b_1h_2}T_2^{a_2h_1+b_2h_2}f\, d\mu. \end{align*} $$

If the vector ${\mathbf {a}}$ repeats s times, we abbreviate it as ${\mathbf {a}}^{\times s}$ , e.g.

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\mathbf{a}}^{\times 2}, {\textbf{b}}^{\times 3}, {\mathbf{c}}}=\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\mathbf{a}}, {\mathbf{a}}, {\textbf{b}}, {\textbf{b}}, {\textbf{b}}, {\mathbf{c}}}. \end{align*} $$

Box seminorms satisfy the following Gowers–Cauchy–Schwarz inequality [Reference Host12, Proposition 2]:

(16) $$ \begin{align} \limsup_{H\to\infty}\mathopen{}\bigg| \mathop{\mathbb{E}}\limits_{{\underline{h}}\in[H]^s}\int \prod_{{\underline{\epsilon}}\in\{0,1\}^s} {\mathcal C}^{|{\underline{\epsilon}}|} T^{{\textbf{b}}_1 \epsilon_1 h_1 + \cdots + {\textbf{b}}_s \epsilon_s h_s}f_{\underline{\epsilon}}\, d\mu\mathclose{}\bigg|\leq \prod_{{\underline{\epsilon}}\in\{0,1\}^s}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f_{\underline{\epsilon}}|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}. \end{align} $$

(One can replace the limsup with a limit since it is known to exist.)

We frequently bound one seminorm in terms of another. An inductive application of equation (12), or alternatively a simple application of the Gowers–Cauchy–Schwarz inequality in equation (16), yields the following monotonicity property:

(17) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}\leq \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots,{\textbf{b}}_s, {\textbf{b}}_{s+1}}, \end{align} $$

a special case of which is the aforementioned bound $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s,T}\leq \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s+1, T}$ for any ${f\in L^{\infty }(\mu )}$ and system $(X, {\mathcal X}, \mu , T)$ .

In many of our arguments, we have to deal simultaneously both with a collection of transformations and their powers. The relevant box seminorms are compared in the following lemma.

Lemma 2.1. [Reference Frantzikinakis and Kuca11, Lemma 3.1]

Let $\ell ,s\in {\mathbb N}$ , $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ be a system, ${f\in L^{\infty }(\mu )}$ be a function, ${\textbf {b}}_1, \ldots , {\textbf {b}}_s\in {\mathbb Z}^{\ell }$ be vectors, and $r_1, \ldots , r_s\in {\mathbb Z}$ be non-zero. Then

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}\leq \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{r_1 {\textbf{b}}_1, \ldots, r_s {\textbf{b}}_s}, \end{align*} $$

and if $s\geq 2$ , we additionally get the bound

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{r_1 {\textbf{b}}_1, \ldots, r_s {\textbf{b}}_s} \leq (r_1\cdots r_s)^{1/2^{s}}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}. \end{align*} $$

The following lemma allows us to compare box seminorms depending on the invariant $\sigma $ -algebras of the transformations involved.

Lemma 2.2. Let $\ell ,s\in {\mathbb N}$ , $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ be a system of commuting transformations, and ${\textbf {b}}_1, \ldots , {\textbf {b}}_s, {\mathbf {c}}_1, \ldots , {\mathbf {c}}_s\in {\mathbb Z}^{\ell }$ be vectors with the property that ${\mathcal I}(T^{{\textbf {b}}_i})\subseteq {\mathcal I}(T^{{\mathbf {c}}_i})$ for each $i\in [s]$ . Then $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_s}\leq \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\mathbf {c}}_1, \ldots , {\mathbf {c}}_s}$ for each $f\in L^{\infty }(\mu )$ .

Proof. We prove this by induction on s. For $s=1$ , we simply have

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1}=\Vert {\mathbb{E}}(f|{\mathcal I}(T^{{\textbf{b}}_1}))\Vert_{L^2(\mu)}\leq \Vert {\mathbb{E}}(f|{\mathcal I}(T^{{\mathbf{c}}_1}))\Vert_{L^2(\mu)} = \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\mathbf{c}}_1}, \end{align*} $$

where we use the fact that $\Vert {\mathbb {E}}(f|{\mathcal A})\Vert _{L^2(\mu )}\leq \Vert {\mathbb {E}}(f|{\mathcal B})\Vert _{L^2(\mu )}$ whenever ${\mathcal A}\subseteq {\mathcal B}$ . For $s>1$ , we use the induction formula for seminorms and the result for $s=1$ to deduce that

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s}^{2^s} &=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in[H]^{s-1}}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{{\textbf{b}}_1, \ldots, {\textbf{b}}_{s-1};{\underline{h}}} f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_s}^{2^{s-1}}\\ &\leq \lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in[H]^{s-1}}\lvert\hspace{-1.2pt}|\hspace{-1.2pt}| \Delta_{{\textbf{b}}_1, \ldots, {\textbf{b}}_{s-1};{\underline{h}}} f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\mathbf{c}}_s}^{2^{s-1}} = \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_{s-1}, {\mathbf{c}}_s}^{2^s}. \end{align*} $$

The claim follows by iterating this procedure $s-1$ more times.

2.3 Dual functions and sequences

Let $s\in {\mathbb N}$ and $\{0,1\}^s_* = \{0,1\}^s\setminus \{\underline {0}\}$ . For a system $(X, {\mathcal X}, \mu , T)$ and $f\in L^{\infty }(\mu ),$ we define

$$ \begin{align*} {\mathcal D}_{s, T}(f) := \lim_{M\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{m}}\in [M]^s}\prod_{{\underline{\epsilon}}\in\{0,1\}^s_*}{\mathcal C}^{|{\underline{\epsilon}}|}T^{{\underline{\epsilon}}\cdot{\underline{m}}}f \end{align*} $$

(the limit exists in $L^2(\mu )$ by [Reference Host and Kra13]). We call ${\mathcal D}_{s,T}(f)$ the dual function of f of level s with respect to T. The name comes because of the identity

(18) $$ \begin{align} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{s, T}^{2^s} = \int f \cdot {\mathcal D}_{s, T}(f)\, d\mu, \end{align} $$

a consequence of which is that the span of dual functions of degree s is dense in $L^1({\mathcal Z}_{s-1}(T))$ .

Let $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ be a system. Using the identities in equations (15) and (18), we get

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}, {\textbf{b}}_{s+1}^{\times s'}}^{2^{s+s'}}=\lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in [H]^s}\int \Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}} f \cdot {\mathcal D}_{s', T^{{\textbf{b}}_{s+1}}}(\Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}}f)\, d\mu, \end{align*} $$

the special case of which is

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{{\textbf{b}}_1}, \ldots, {\textbf{b}}_{s+1}}^{2^{s+1}} = \lim_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}}\in [H]^s} \int \Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}}f \cdot {\mathbb{E}}(\Delta_{{{\textbf{b}}_1}, \ldots, {{\textbf{b}}_{s}}; {\underline{h}}}\overline{f}|{\mathcal I}(T^{{\textbf{b}}_{s+1}}))\, d\mu. \end{align*} $$

For $s\in {\mathbb N}$ , we denote

$$ \begin{align*} {\mathfrak D}_s := \{(T_j^n {\mathcal D}_{s', T_j}f)_{n\in{\mathbb Z}}: f\in L^{\infty}(\mu),\, j\in[\ell],\, 1\leq s'\leq s\} \end{align*} $$

to be the set of sequences of 1-bounded functions coming from dual functions of degree up to s for the transformations $T_1,\ldots , T_{\ell }$ , and moreover we define ${\mathfrak D} := \bigcup _{s\in {\mathbb N}}{\mathfrak D}_s$ .

The utility of dual functions comes from the following approximation result.

Proposition 2.3 will be used as follows. Suppose that the $L^2(\mu )$ limit of the average ${\mathbb {E}}_{n\in [N]}\, T_1^{p_1(n)}f_1 \cdots T_{\ell }^{p_{\ell }(n)}f_{\ell }$ vanishes whenever $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_{\ell }|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T_{\ell }} = 0$ . If

$$ \begin{align*} \limsup_{N\to\infty}\Big\Vert \mathop{\mathbb{E}}\limits_{n\in[N]}\, T_1^{p_1(n)}f_1 \cdots T_{\ell}^{p_{\ell}(n)}f_{\ell}\Big\Vert_{L^2(\mu)}>0 \end{align*} $$

for some functions $f_1, \ldots , f_{\ell }\in L^{\infty }(\mu )$ , then we decompose $f_{\ell }$ as in Proposition 2.3 for sufficiently small $\varepsilon>0$ so that

$$ \begin{align*} \limsup_{N\to\infty}\bigg\Vert \mathop{\mathbb{E}}\limits_{n\in[N]}\prod_{j\in[\ell-1]}T_j^{p_j(n)}f_j \cdot \sum_k c_k T_{\ell}^{p_{\ell}(n)}{\mathcal D}_{s, T_{\ell}}(g_k)\bigg\Vert_{L^2(\mu)}>0 \end{align*} $$

for some (finite) linear combinations of dual functions. Applying the triangle inequality and the pigeonhole principle, we deduce that there exists k for which

$$ \begin{align*} \limsup_{N\to\infty}\bigg\Vert \mathop{\mathbb{E}}\limits_{n\in[N]}\prod_{j\in[\ell-1]}T_j^{p_j(n)}f_j \cdot {\mathcal D}(p_{\ell}(n))\bigg\Vert_{L^2(\mu)}>0, \end{align*} $$

where ${\mathcal D}(n)(x) := T_{\ell }^{n}{\mathcal D}_{s, T_{\ell }}g_k(x)$ for $n\in {\mathbb N}$ and $x\in X$ . This way, we essentially replace the term $T_{\ell }^{p_{\ell }(n)}f_{\ell }$ in the original average by the more structured piece ${\mathcal D}(p_{\ell }(n))$ .

2.4 Eigenfunctions and criterion for weak joint ergodicity

Following [Reference Frantzikinakis and Host10], we define the notion of eigenfunctions that appears in the statement of Theorem 1.2.

We denote the set of non-ergodic eigenfunctions with respect to T by ${\mathcal E}(T)$ . For ergodic systems, a non-ergodic eigenfunction is either the zero function or a classical unit modulus eigenfunction. For general systems, each function $\chi \in {\mathcal E}(T)$ satisfies $\chi (Tx)\,{=}\,\mathbf {1}_E(x) \, e(\phi (x))\, \chi (x)$ for some T-invariant set $E\in {\mathcal X}$ and measurable T-invariant function $\phi \colon X\to {\mathbb T}$ .

Definition. (Weak joint ergodicity)

We say that a collection of sequences $a_1, \ldots , a_{\ell }: {\mathbb N}\to {\mathbb Z}$ is weakly jointly ergodic for the system $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ , if

$$ \begin{align*} \lim_{N\to\infty}\bigg\Vert \frac{1}{N}\sum_{n=1}^N T_1^{a_1(n)}f_1 \cdots T_{\ell}^{a_{\ell}(n)}f_{\ell} - {\mathbb{E}}(f_1|{\mathcal I}(T_1)) \cdots {\mathbb{E}}(f_{\ell}|{\mathcal I}(T_{\ell}))\bigg\Vert_{L^2(\mu)} = 0 \end{align*} $$

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

The notion of non-ergodic eigenfunction is important for us because of the following criterion for weak joint ergodicity from [Reference Frantzikinakis and Kuca11].

When $T_1, \ldots , T_{\ell }$ are ergodic, the condition in equation (19) can be restated as follows:

$$ \begin{align*} \lim_{N\to\infty}\mathop{\mathbb{E}}\limits_{n\in[N]} \, e(\alpha_1 a_1(n) + \cdots + \alpha_{\ell} a_{\ell}(n)) = 0 \end{align*} $$

for all $\alpha _j\in \operatorname {\mathrm {Spec}}(T_j)$ , $j\in [\ell ]$ , not all zero. Here,

$$ \begin{align*}\operatorname{\mathrm{Spec}}(T) := \{\alpha\in{\mathbb T}: T\chi = e(\alpha) \chi \textrm{ for some } \chi\in{\mathcal E}(T)\}.\end{align*} $$

We will apply Theorem 2.4 in the proof of Theorem 1.2. The first condition will be satisfied thanks to the stronger result proved in Theorem 1.1.

5.1 The formalism behind the induction scheme

We start by introducing a handy formalism used for the induction scheme in the proof of Theorem 1.1. We often associate the average from equation (35) with the tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ . This tuple does not contain any information about the polynomials $q_1, \ldots , q_L$ , but this is not necessary. These terms play no role in our inductive argument and can be easily disposed of using Proposition 3.7. The only thing they do influence is the degree s of the seminorm with which we end up controlling the average from equation (35).

Definition. (Indexing data)

For an average from equation (35) or the associated tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ , we let $\ell $ be its length, $d:= \max \nolimits _{j\in [\ell ]}\deg \rho _j$ be its degree, and $\eta $ be its indexing tuple. For $j\in [\ell ]$ , we set $d_j := \deg \rho _j$ . We let $K_1$ be the maximum number of pairwise independent polynomials within the family $\rho _1,\ldots , \rho _{\ell }$ (we set $K_1:=1$ if $\ell =1$ or every two polynomials are pairwise dependent). We partition $[\ell ]=\bigcup _{t\in [K_1]} {\mathfrak I}_t$ , where $j_1, j_2$ belong to the same ${\mathfrak I}_t$ if and only if $\rho _{j_1}$ , $\rho _{j_2}$ are linearly dependent. Thus, ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ partitions $[\ell ]$ into index sets corresponding to families of pairwise dependent polynomials. Furthermore, we define

$$ \begin{align*} {\mathfrak L} := \{j\in[\ell]: \deg \rho_j = d\} \end{align*} $$

to be the set of indices corresponding to polynomials of maximum degree, and we rearrange ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ so that ${\mathfrak L} =\bigcup _{t\in [K_2]}{\mathfrak I}_t$ for some $K_2\leq K_1$ . We also let $K_3 := |{\mathfrak L}|$ be the number of maximum degree polynomials, and we notice that $K_2\leq K_3\leq \ell $ .

Sometimes, we denote ${\mathfrak L} = {\mathfrak L}(\rho _1, \ldots , \rho _{\ell }), {\mathfrak I}_t={\mathfrak I}_t(\rho _1, \ldots , \rho _{\ell })$ , and $K_i = K_i (\rho _1, \ldots , \rho _{\ell })$ to emphasize the dependence on a specific family of polynomials.

Example 3. Consider the tuple

(36) $$ \begin{align} (T_1^{n^2}, T_2^{n^2}, T_3^{n^2+n}, T_4^{2n^2+2n}, T_3^{n^2+2n}, T_6^n, T_2^{n^2+3n}). \end{align} $$

It has length 7, degree 2, indexing tuple $\eta = (1, 2, 3, 4, 3, 6, 2)$ , $K_1 = 5$ (corresponding to five pairwise independent polynomials $n^2, n^2+n, n^2 + 2n, n^2 + 3n, n$ ), $K_2 = 4$ (corresponding to four quadratic pairwise independent polynomials $n^2, n^2 + n, n^2 + 2n, n^2+3n$ ), $K_3 = 6$ (corresponding to six quadratic polynomials appearing in equation (36)), ${\mathfrak L} = \{1, 2, 3, 4, 5, 7\}$ , and partition ${\mathfrak I}_1 = \{1, 2\}, {\mathfrak I}_2 = \{3, 4\}, {\mathfrak I}_3 = \{5\}, {\mathfrak I}_4 = \{7\}, {\mathfrak I}_5 = \{6\}$ .

The rationale behind introducing the partition ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ and the indexing tuple $\eta $ is as follows. As part of our assumptions, we know that $T_i^{\beta _i} T_j^{-\beta _j}$ is ergodic whenever the polynomials $\rho _i, \rho _j$ are dependent, $b_i, b_j$ are their leading coefficients, and $\beta _i := b_i/\gcd (b_i, b_j), \beta _j := b_j/\gcd (b_i, b_j)$ . Introducing the partition ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ allows us to keep track of pairs $(i,j)$ for which we have these ergodicity properties. The reason for introducing the indexing tuple $\eta $ is that in our induction procedure, we gradually replace a transformation $T_{\eta _m}$ in the tuple $(T_{\eta _j}^{\,\rho _j(n)})_{j\in [\ell ]}$ by a different transformation $T_{\eta _i}$ , and so $\eta $ keeps track of these substitutions.

Definition. (Good ergodicity property along $\eta $ )

Let $\eta \in [\ell ]^{\ell }$ be an indexing tuple and $b_1, \ldots , b_{\ell }$ be the leading coefficients of the polynomials $\rho _1, \ldots , \rho _{\ell }$ . We say that the system $(X,{\mathcal X}, \mu ,T_1, \ldots , T_{\ell })$ has the good ergodicity property along $\eta $ for the polynomials $\rho _1, \ldots , \rho _{\ell }$ if for every distinct $\eta _{j_1}, \eta _{j_2}$ belonging to the same ${\mathfrak I}_t$ with $t\in [K_1]$ , we have

(37) $$ \begin{align} {\mathcal I}(T_{\eta_{j_1}}^{\beta_{j_1}}T_{\eta_{j_2}}^{-\beta_{j_2}}) = {\mathcal I}(T_{\eta_{j_1}})\cap {\mathcal I}(T_{\eta_{j_2}}), \end{align} $$

where $\beta _{j} := b_{j}/\gcd (b_{j_1}, b_{j_2})$ for $j=j_1, j_2$ . In other words, the only functions invariant under $T_{\eta _{j_1}}^{\beta _{j_1}}T_{\eta _{j_2}}^{-\beta _{j_2}}$ are those simultaneously invariant under $T_{\eta _{j_1}}$ and $T_{\eta _{j_2}}$ . In particular, having the good ergodicity property corresponds to having the good ergodicity property for the polynomials $p_1,\ldots , p_{\ell }$ along the identity indexing tuple $\eta _0 := (1, \ldots , \ell )$ . We similarly say that the tuple $(T_{\eta _j}^{\,\rho _j(n)})_{j\in [\ell ]}$ has the good ergodicity property if $(X, {\mathcal X}, \mu , T_1, \ldots , T_{\ell })$ has the good ergodicity property along $\eta $ for $\rho _1, \ldots , \rho _{\ell }$ .

What this definition captures is that every time we encounter in our tuple two transformations $T_{\eta _{j_1}}, T_{\eta _{j_2}}$ whose indices $\eta _{j_1}, \eta _{j_2}$ lie in the same set ${\mathfrak I}_t$ , the identity in equation (37) is satisfied. For instance, the tuple in equation (36) has the good ergodicity property along $\eta $ precisely when

$$ \begin{align*} {\mathcal I}(T_1 T_2^{-1}) = {\mathcal I}(T_1)\cap{\mathcal I}(T_2)\quad \textrm{and}\quad {\mathcal I}(T_3 T_4^{-2}) = {\mathcal I}(T_3)\cap {\mathcal I}(T_4). \end{align*} $$

The first identity corresponds to comparing the pairs $(j_1, j_2) = (1,2), (1,7)$ corresponding to the occurrences of transformations $T_1$ and $T_2$ from the cell ${\mathfrak I}_1$ ( $T_1$ occurs at the index 1 and $T_2$ occurs at the indices 2 and 7). The second identity comes from comparing the pairs $(j_1, j_2) = (3, 4), (4, 5)$ corresponding to the transformations $T_3, T_4$ from the cell ${\mathfrak I}_2$ ( $T_3$ occurs at the indices 3 and 5 whereas $T_4$ occurs at the index 4).

The guiding principle behind our arguments is that we derive seminorm control of the average from equation (35) by inductively applying seminorm control of an average that is ‘simpler’ than the original average in an appropriate sense. For instance, in Proposition 4.2 and Corollary 4.5, we obtained seminorm control for the tuple $(T_1^{n^2}, T_2^{n^2}, T_3^{n^2+n})$ from Example 1 by invoking seminorm control for the following tuples:

• • $(T_1^{n^2}, T_2^{n^2}, T_2^{n^2+n})$ in the ping step of the smoothing argument in Proposition 4.2;

• • $(T_1^{n^2}, *, T_3^{n^2+n})$ in the pong step of the smoothing argument in Proposition 4.2 (the asterisk is introduced purely for convenience; it denotes the term replaced by a product of dual functions);

• • $(T_1^{n^2}, T_2^{n^2}, *)$ in Corollary 4.5.

The relative complexity of a tuple or an average is captured by the following notion.

Definition. (Type)

The type of $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ is the tuple $w = (w_1, \ldots , w_{K_2})$ , where

$$ \begin{align*} w_t := |\{j\in {\mathfrak L}: {\eta_j}\in {\mathfrak I}_t\}| = |\{j\in[\ell]: \deg \rho_j = d,\; {\eta_j}\in {\mathfrak I}_t\}| \end{align*} $$

counts the number of times the transformations $(T_j)_{j\in {\mathfrak I}_t}$ appear in $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ with a polynomial iterate of maximum degree. (We note here that the type of $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ depends not just on $\eta $ and the polynomials $\rho _1, \ldots , \rho _{\ell }$ , but also on the ordering of the sets ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ . We do not record this dependence explicitly, instead fixing some ordering of ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ a priori.) We note that $|w| := w_1 +\cdots + w_{K_2} = K_3$ . We say that the type w is basic if it has the form $w=(K_3,0,\ldots , 0)$ .

For instance, the tuple in equation (36) has type $(3,3,0,0)$ : this is because $T_1, T_2$ corresponding to ${\mathfrak I}_1$ occur thrice, as do the transformations $T_3, T_4$ corresponding to ${\mathfrak I}_2$ , while the transformations $T_5, T_7$ corresponding to ${\mathfrak I}_3$ and ${\mathfrak I}_4$ do not occur at all. We do not care about the occurrence of $T_6$ since it has a linear iterate.

It is instructive to see what happens when the polynomials $\rho _1, \ldots , \rho _{\ell }$ are pairwise independent. In that case, ${\mathfrak I}_t = \{j_t\}$ for every $t\in [\ell ]$ and $w_t$ counts the number of times the transformation $T_{j_t}$ appears among $(T_{\eta _j})_{j\in {\mathfrak L}}$ , or equivalently the number of times that $T_{j_t}$ attains a polynomial iterate of maximal degree. So for pairwise independence polynomials, this notion of type recovers the concept of type from [Reference Frantzikinakis and Kuca11, §8.2] (up to permuting ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ ).

For the set of tuples in ${\mathbb N}_0^{K_2}$ of length $K_3$ , we say $w'<w$ if there exists ${\kappa \in [K_2-1]}$ such that for all $t\in [\kappa ]$ , we have $w^{\prime }_t = w_t$ , and either $w^{\prime }_{\kappa +1} = 0 < w_{\kappa +1}$ or $w^{\prime }_{\kappa +1}>w_{\kappa +1}>0$ . For instance, we have the following chain of type inequalities:

$$ \begin{align*} (4, 0, 0) &< (3, 0, 1) < (3, 1, 0) < (2, 0, 2) < (2, 2, 0)\\ &< (2, 1, 1) < (1, 0, 3) < (1, 2, 1) < (1, 1, 2). \end{align*} $$

The first, third, fifth, sixth, and eighth inequalities follow from the condition $w^{\prime }_{\kappa +1}>w_{\kappa +1}>0$ while the second, fourth, and seventh inequalities are consequences of the condition $w^{\prime }_{\kappa +1} = 0 < w_{\kappa +1}$ . This is a rather atypical ordering, but it turns out to determine well which of the tuples $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ , $\mathopen {}( T_{\eta ^{\prime }_j}^{\,\rho ^{\prime }_j(n)} \mathclose {})_{j\in [\ell ]}$ is ‘simpler’ than the other. The motivation for this particular choice of ordering is that in the arguments to come, we will be passing from a tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ of type w to another tuple $\mathopen {}( T_{\eta ^{\prime }_j}^{\,\rho ^{\prime }_j(n)} \mathclose {})_{j\in [\ell ]}$ of type $w'<w$ in two ways. In one of them, the type $w'$ will meet the condition $w^{\prime }_{\kappa +1}>w_{\kappa +1}>0$ while in the other one, it will satisfy the condition $w^{\prime }_{\kappa +1} = 0 < w_{\kappa +1}$ . Arguing this way, we arrive in finitely many steps at a tuple of a basic type $w=(K_3, 0, \ldots , 0)$ , which constitutes the base case of our induction. This transition will be explained in greater detail at the end of §5.2 and illustrated in Example 10.

5.2 The general strategy

In this section, we outline how to obtain a seminorm control of a given tuple using seminorm control for tuples of lower type or shorter length.

The previous notions of controllability are supposed to capture whether Proposition 3.7 is applicable to the relevant tuples in our setting.

Example 4. (Controllable versus uncontrollable tuples)

Consider the following two tuples:

(38) $$ \begin{align} &\mathopen{}( T_1^{n^2}, T_2^{n^2}, T_3^{n^2}, T_4^{n^2}, T_5^{n^2+n}, T_1^{n^2+n}, T_5^{n^2+2n}, T_5^{n^2+2n} \mathclose{}), \end{align} $$

(39) $$ \begin{align} &\mathopen{}( T_1^{n^2}, T_2^{n^2}, T_3^{n^2}, T_4^{n^2}, T_1^{n^2+n}, T_1^{n^2+n}, T_5^{n^2+2n}, T_5^{n^2+2n} \mathclose{}). \end{align} $$

Defining the partitions ${\mathcal I}_1 = \{1, 2, 3, 4\}$ , ${\mathcal I}_2 = \{5, 6\}$ , ${\mathcal I}_3 = \{7, 8\}$ corresponding to the independent polynomials $n^2, n^2+n, n^2+2n$ . respectively, the first tuple has type $(5, 3, 0)$ while the second one has type $(6, 2, 0)$ , and for both tuples, we have $t_w=2$ . The first one is controllable because for the index $m = 5$ , the only values $i\neq m$ such that $\eta _i = 5$ are $i=7,8$ corresponding to the polynomial $n^2+2n$ , which is distinct from $n^2+n$ .

By contrast, the tuple in equation (39) does not possess an index satisfying the controllability condition: the only indices $m\in [8]$ with $\eta _m\in {\mathfrak I}_2$ are $m=7,8$ , and we have both $\eta _7 = \eta _8$ and $\rho _7(n) = n^2+2n = \rho _8(n)$ . Hence, this tuple is uncontrollable.

Our strategy for proving seminorm control will work rather differently for controllable and uncontrollable tuples. Suppose first that the average from equation (35) with tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ of a non-basic type is controllable, and that an index m satisfies the controllability condition. Then Proposition 3.7 guarantees that the average from equation (35) is controlled by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_m|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s+1}}$ for some vectors ${\textbf {b}}_1, \ldots , {\textbf {b}}_{s+1}$ from equation (23), and the controllability implies that these vectors are indeed non-zero. We want to show that this average is also controlled by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_m|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s}, {\mathbf {e}}_{\eta _m}^{\times s'}}$ for some $s'\in {\mathbb N}$ via a seminorm smoothing argument that generalizes Proposition 4.2. We then iterate this result s more times to get control by a $T_{\eta _m}$ -seminorm of $f_m$ . The seminorm smoothing argument follows a ping-pong strategy much like in the proof of Proposition 4.2. We first show that control by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_m|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s+1}}$ implies control by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_i|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s}, {\mathbf {e}}_{\eta _i}^{\times s_1}}$ for some $s_1\in {\mathbb N}$ and $i\neq m$ , and then we use this auxiliary result to obtain control by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_m|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s}, {\mathbf {e}}_{\eta _m}^{\times s'}}$ .

The main idea behind the ping step of the seminorm smoothing argument is to show that a seminorm control of the average from equation (35) can be deduced from a seminorm control of a family of averages of the form

(40) $$ \begin{align} \lim_{N\to\infty}\mathop{\mathbb{E}}\limits_{n\in [N]} \,\prod_{j\in[\ell]}T_{{\eta_j'}}^{\,\rho^{\prime}_j(n)}f^{\prime}_j \cdot \prod_{j\in[L']}{\mathcal D}^{\prime}_{j}(q^{\prime}_j(n)) \end{align} $$

for some polynomials $\rho ^{\prime }_1, \ldots , \rho ^{\prime }_{\ell }, q^{\prime }_1, \ldots , q^{\prime }_{L'}\in {\mathbb Z}[n]$ , 1-bounded functions $f^{\prime }_1, \ldots , f^{\prime }_{\ell }\in L^{\infty }(\mu )$ , and sequences of functions ${\mathcal D}^{\prime }_1, \ldots , {\mathcal D}^{\prime }_{L'}\in {\mathfrak D}$ . Moreover, the indexing tuple $\eta '\in [\ell ]^{\ell }$ is obtained from $\eta $ by changing $\eta _m$ into $\eta _i$ for some $i\neq m$ , that is, the passage from equation (35) to equation (40) goes by replacing $T_{\eta _m}$ at index m with $T_{\eta _i}$ . Importantly, the new average from equation (40) satisfies several key properties:

1. (i) it has a lower type than the original average, so that we can argue by induction;

2. (ii) the new average retains the good ergodicity property of the original average;

3. (iii) the functions $f^{\prime }_j$ in the new average satisfy some invariance properties;

4. (iv) as long as the aforementioned invariance properties are satisfied, the new average from equation (40) is controlled by the seminorm $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_j'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T_{\eta ^{\prime }_j}}$ for each $j\in [\ell ]$ and some $s\in {\mathbb N}$ .

Proposition 6.1 explains the exact way in which we pass from averages from equation (35) to those from equation (40) so that the property (i) is satisfied, and Proposition 6.4 establishes the property (ii). Proposition 6.5 then ensures that the functions $f_j'$ in equation (40) satisfy needed invariance properties.

We note though that the new average from equation (40) need not be controllable. For instance, if we take the average corresponding to the tuple from equation (38), then in the ping step, we replace $T_5^{n^2+n}$ by the same iterate of one of $T_1, T_2, T_3, T_4$ . The new average is then uncontrollable, as is the tuple from equation (39), corresponding to replacing $T_5^{n^2+n}$ by $T_1^{n^2+n}$ . Hence, controllability may not be preserved while performing the procedure outlined above.

In the pong step of the smoothing argument for equation (35), we deal with averages of the form

(41) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in [N]} \,\prod_{\substack{j\in[\ell],\\ j\neq i}}T_{\eta_j}^{\,\rho_j(n)}f^{\prime\prime}_j \cdot \prod_{j\in[L"]}{\mathcal D}^{\prime\prime}_{j}(q^{\prime\prime}_j(n)). \end{align} $$

Crucially, each function $f^{\prime \prime }_j$ is invariant under some composition of $T_{\eta _j}$ and $T_j$ . This allows us to replace (some iterate of) $T_{\eta _j}$ in equation (41) by (some iterate of) $T_j$ , a procedure that we call flipping, and show that an average from equation (41) essentially equals an average of the form

(42) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in [N]} \,\prod_{\substack{j\in[\ell],\\ j\neq i}}T_j^{\,\rho^{\prime\prime\prime}_j(n)}f^{\prime\prime\prime}_j \cdot \prod_{j\in[L"]}{\mathcal D}^{\prime\prime}_{j}(q^{\prime\prime\prime}_j(n)) \end{align} $$

of length $\ell -1$ . The details of how flipping is performed are presented in Proposition 6.6. An inductive application of a suitable modification of Theorem 1.1 then gives a control of equation (42) by a $T_j$ -seminorm of $f^{\prime \prime \prime }_j$ for each $j\neq i$ , and the invariance property of $f^{\prime \prime }_j$ translates it into a control of equation (41) by a $T_{\eta _j}$ -seminorm of $f^{\prime \prime }_j$ for each $j\neq i$ . A straightforward argument analogous to one at the end of the proof of Proposition 4.2 gives a control of equation (35) by a $T_{\eta _m}$ -seminorm of $f_m$ .

If the average from equation (35) is uncontrollable, then we proceed rather differently. The previous strategy breaks right at the start since there is no index m satisfying the controllability condition. Consequently, whichever index m with $\eta _m\in {\mathfrak I}_{t_w}$ we take, we cannot employ Proposition 3.7 to bound the seminorm by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_m|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_{s+1}}$ for non-zero vectors ${\textbf {b}}_1, \ldots , {\textbf {b}}_{s+1}$ . What we use instead is the inductive assumption that the functions $f_j$ at indices j with $\eta _j\in {\mathfrak I}_{t_w}$ are invariant under a composition of (some power of) $T_{\eta _j}$ and (some power of) $T_j^{-1}$ . Using this invariance property, we perform flipping once more to replace the original average from equation (35) by a new average

(43) $$ \begin{align} \lim_{N\to\infty}\mathop{\mathbb{E}}\limits_{n\in [N]} \,\prod_{j\in[\ell]}T_{{\eta_j""}}^{\,\rho^{\prime\prime\prime\prime}_j(n)}f^{\prime\prime\prime\prime}_j \cdot \prod_{j\in[L]}{\mathcal D}_{j}(q^{\prime\prime\prime\prime}_j(n)), \end{align} $$

where $\eta ^{\prime \prime \prime \prime }_j = j$ whenever $\eta _j\in {\mathfrak I}_{t_w}$ ; the details are provided in Corollary 6.7. This new average has the good ergodicity property and is controllable. Importantly, it has a lower type, which is established in Proposition 6.8. We can then obtain seminorm control of equation (35) by inductively invoking the seminorm control of equation (43). The seminorm control of equation (43) is proved in turn by the smoothing argument for controllable averages described above.

Thus, whether the tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ of a non-basic type is controllable or not, the idea is to control it by a Gowers–Host–Kra seminorm by invoking seminorm control for tuples of lower type or smaller length that naturally appear when examining $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ . If the tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ of type w is controllable, we will invoke seminorm control for tuples of type $w'$ satisfying $w^{\prime }_t = w_t$ for $t\in [\kappa ]$ and $w^{\prime }_{\kappa +1}>w_{\kappa +1}>0$ for some $\kappa \in [K_2-1]$ in the ping step of the smoothing argument. Specifically, the new type $w'$ is obtained from w by the type operation defined in equation (45). In the pong step of the smoothing argument, we will use seminorm control for tuples of length $\ell -1$ . If the tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ is uncontrollable, we will invoke seminorm control for tuples of type $w'$ satisfying $w^{\prime }_t = w_t$ for $t\in [\kappa ]$ and $w^{\prime }_{\kappa +1}=0<w_{\kappa +1}$ for some $\kappa \in [K_2-1]$ ; the details are given in Proposition 6.8(v). The way in which we apply seminorm control for tuples of lower type motivates the choice of our somewhat weird ordering on types.

Reducing to tuples of lower type this way and noting that the tuples of length $\ell $ can have at most $(\ell +1)^{\ell }$ distinct types, we arrive after finitely many steps at tuples of basic type $w = (K_3, 0, \ldots , 0)$ , that is, those in which all the transformations come from the same class ${\mathfrak I}_1$ . Tuples of basic type will serve as the basis for our induction procedure. For instance, the tuple $(T_1^{n^2}, T_2^{n^2}, T_2^{n^2+n})$ from Example 1 has basic type $(3, 0)$ because it only involves the transformations $T_1, T_2$ whose indices belong to the set ${\mathfrak I}_1 = \{1, 2\}$ (corresponding to the polynomial $n^2$ ); however, the type $(2, 1)$ of $(T_1^{n^2}, T_2^{n^2}, T_3^{n^2+n})$ is not basic because this tuple involves both the transformations $T_1, T_2$ and the transformation $T_3$ with an index from the set ${\mathfrak I}_2 = \{3\}$ (corresponding to the polynomial $n^2 + n$ ).

6.1 Reducing controllable tuples to tuples of lower type in the ping step

As explained in §5.2, in the ping part of the smoothing argument, we will replace the original tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ by a new tuple $\mathopen {}( T_{\eta ^{\prime }_j}^{\,\rho ^{\prime }_j(n)} \mathclose {})_{j\in [\ell ]}$ . The new indexing tuple $\eta '$ will be defined via the operation

(44) $$ \begin{align} (\tau_{mi}\eta)_j := \begin{cases} \eta_j, &j\neq m,\\ \eta_i, &j = m, \end{cases} \end{align} $$

for some distinct values $m, i\in {\mathfrak L}$ . This indexing tuple corresponds to replacing the term $T_{\eta _m}^{\,\rho _m(n)}$ in $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ by $T_{\eta _i}^{\,\rho ^{\prime }_i(n)}$ , and all the other terms $T_{\eta _j}^{\,\rho _j(n)}$ by $T_{\eta _j}^{\,\rho ^{\prime }_j(n)}$ . The new tuple $\mathopen {}( T_{\eta ^{\prime }_j}^{\,\rho ^{\prime }_j(n)} \mathclose {})_{j\in [\ell ]}$ has to be chosen carefully: it must preserve the good ergodicity property and allow for seminorm control. Lastly, it must have a lower type. For this reason, we let $\operatorname {\mathrm {Supp}}(w):= \{t\in [K_2]: w_t>0\}$ , and if there exist distinct integers $t_1,t_2\in \operatorname {\mathrm {Supp}}(w)$ , we define the type operation $\sigma _{t_1 t_2}w$ by the formula

(45) $$ \begin{align} (\sigma_{t_1 t_2}w)_t := \begin{cases} w_t, &t \neq t_1, t_2,\\ w_{t_1}-1, &t = t_1,\\ w_{t_2} + 1, &t = t_2. \end{cases}. \end{align} $$

For instance, $\sigma _{32}(3, 2, 2) = (3, 3, 1)$ . As a consequence of our ordering on types, we have $\sigma _{t_1 t_2}w < w$ whenever $t_2 < t_1$ (the assumption $w_{t_2}>0$ is crucial here), so in particular $(3, 3, 1) < (3, 2, 2)$ .

Proposition 6.1, which we are about to state now, specifies how these tuples of lower type are picked, what form they take, and what properties they enjoy. It will be used in our smoothing argument in Proposition 7.5 in that the tuple of lower type for which we invoke the induction hypothesis in the ping step is constructed in Proposition 6.1.

We remark that when the leading coefficients of $\rho _1, \ldots , \rho _{\ell }$ are 1, then $\rho ^{\prime }_j = \rho _j$ for every $j\in [\ell ]$ , so the property (iii) becomes trivial.

Proof. Let $t_w$ be the last non-zero index of the type w of $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ . By the controllability of the tuple, there exists $m\in [\ell ]$ with $\eta _m\in {\mathfrak I}_{t_w}$ such that $\eta _{m'}=\eta _m$ implies that $\rho _{m'}$ and $\rho _m$ are distinct. Let $t^{\prime }_w\in \operatorname {\mathrm {Supp}}(w)$ be an index different from $t_w$ (it exists since the type w is non-basic) and $i\in [\ell ]$ be an index with $\eta _i\in {\mathfrak I}_{t^{\prime }_w}$ . We define $\eta ':=\tau _{mi}\eta $ , meaning that we replace $T_{\eta _m}$ by $T_{\eta _i}$ and keep the other transformations the same.

We let $\unicode{x3bb} \in {\mathbb N}$ be the smallest number for which $({\unicode{x3bb} }/{b_m})\rho _m\in {\mathbb Z}[n]$ (equivalently, $\unicode{x3bb} $ is the smallest number such that $b_m$ divides the coefficients of the polynomials $\rho _m(\unicode{x3bb} n +~r) - \rho _m(r)$ for $r\in {\mathbb Z}$ ). We also fix an arbitrary $r\in \{0, \ldots , \unicode{x3bb} -1\}$ . We then define the new polynomials $\rho ^{\prime }_1, \ldots , \rho ^{\prime }_{\ell }$ by the formula

$$ \begin{align*} \rho^{\prime}_{j}(n) := \begin{cases} \rho_{j}(\unicode{x3bb} n + r) - \rho_{j}(r), &j\neq m,\\[4pt] \dfrac{b_{i}}{b_{m}}(\rho_j(\unicode{x3bb} n + r) - \rho_j(r)), &j= m. \end{cases} \end{align*} $$

The new polynomials are in ${\mathbb Z}[n]$ by the choice of $\unicode{x3bb} $ , and it is not hard to check that they have zero constant terms. Lastly, the new tuple $\mathopen {}( T_{\eta ^{\prime }_j}^{\,\rho ^{\prime }_j(n)} \mathclose {})_{j\in [\ell ]}$ has the type $w' = \sigma _{t_w t^{\prime }_w}w$ , which is strictly smaller than w by the assumption that $t^{\prime }_w<t_w$ (which follows from ${t^{\prime }_w\neq t_w}$ and the assumption that $t_w$ is the last non-zero index).

Descendancy enjoys the following transitivity property.

Proof. Let $\unicode{x3bb} \in {\mathbb N}$ and $r\in \{0, \ldots , \unicode{x3bb} -1\}$ be such that $\rho _j(n) = {a_{\eta _j}}/{a_j}(p_j(\unicode{x3bb} n+r) - p_j(r))$ . Then a direct computation gives that

$$ \begin{align*} \rho^{\prime}_j(n) = \frac{a_{\eta^{\prime}_j}}{a_j}(p_j(\unicode{x3bb}\unicode{x3bb}' n + \unicode{x3bb} r' + r) - p_j(\unicode{x3bb} r' + r)), \end{align*} $$

giving the claim.

In particular, we get the following corollary of interest to us that follows from a straightforward combination of Proposition 6.1 and Lemma 6.2.

Proof. Suppose that $\rho _1, \ldots , \rho _{\ell }$ are descendants of $p_1, \ldots , p_{\ell }$ along $\eta $ by assumption. Letting $a_j, b_j$ be the leading coefficients of $p_j, \rho _j$ , respectively, and $\rho ^{\prime }_j$ be as defined in Proposition 6.1, we have $b_j = a_{\eta _j} (\unicode{x3bb} ')^{d_j}$ for some $\unicode{x3bb} \in {\mathbb Z}$ , where $d_j := \deg p_j = \deg \rho _j = \deg \rho ^{\prime }_j$ . Thus,

$$ \begin{align*}\rho^{\prime}_j(n) = \rho_j(\unicode{x3bb} n + r) - \rho_j(r) =\frac{a_{\eta^{\prime}_j}}{a_{\eta_j}}(\rho_j(\unicode{x3bb} n + r) - \rho_j(r))\end{align*} $$

for $j\neq m$ and

$$ \begin{align*}\rho^{\prime}_j(n) &= \frac{b_i}{b_m}(\rho_j(\unicode{x3bb} n + r) - \rho_j(r))\\ &= \frac{a_{\eta_i}}{a_{\eta_m}}(\rho_j(\unicode{x3bb} n + r) - \rho_j(r)) = \frac{a_{\eta_m'}}{a_{\eta_m}}(\rho_j(\unicode{x3bb} n + r) - \rho_j(r))\end{align*} $$

for $j = m$ , where we use $\eta ^{\prime }_m = \eta _i$ and $\eta ^{\prime }_j = \eta _j$ for $j\neq m$ . Hence, the polynomials $\rho ^{\prime }_1, \ldots , \rho ^{\prime }_{\ell }$ satisfy the condition of Lemma 6.2, implying the claim.

Descendant tuples enjoy the following important properties.

Property (i) ensures that when passing to descendants, we do not need to redefine the partition ${\mathfrak I}_1, \ldots , {\mathfrak I}_{K_1}$ . Property (ii) is crucial because it shows that descendants retain the essential ergodicity properties of the original tuple.

Proof. Part (i) follows from the fact that for every $j\in [\ell ]$ , the polynomials $p_j$ and $\rho _j$ have the same degree, and that $p_{j_1}, p_{j_2}$ are linearly dependent if and only if $\rho _{j_1}, \rho _{j_2}$ are. We therefore move on to proving part (ii). Let $b_j$ be the leading coefficient of $\rho _j$ , $a_j$ be the leading coefficient of $p_j$ , and $d_j:= \deg p_j = \deg \rho _j$ for every $j\in [\ell ]$ . To check that the tuple $\mathopen {}( T_{\eta _j}^{\,\rho _j(n)} \mathclose {})_{j\in [\ell ]}$ has the good ergodicity property, we need to show that if $\eta _{j_1}, \eta _{j_2}$ are distinct elements of the same set ${\mathfrak I}_{t}$ , then

(46) $$ \begin{align} {\mathcal I}(T_{\eta_{j_1}}^{\beta_{j_1}} T_{\eta_{j_2}}^{-\beta_{j_2}})={\mathcal I}(T_{\eta_{j_1}})\cap{\mathcal I}(T_{\eta_{j_2}}), \end{align} $$

where

$$ \begin{align*} \beta_{j} := b_{j}/\gcd(b_{j_1}, b_{j_2}) \end{align*} $$

for $j = j_1, j_2$ . By construction, $b_j = a_{\eta _j}\unicode{x3bb} ^{d_j}$ for some $\unicode{x3bb} \in {\mathbb N}$ , and so

(47) $$ \begin{align} \beta_{j} = a_{\eta_j}/\gcd(a_{\eta_{j_1}}, a_{\eta_{j_2}})=:\alpha_{\eta_j} \end{align} $$

for $j= j_1, j_2$ . The assumption $\eta _{j_1}, \eta _{j_2}\in {\mathfrak I}_t$ for some fixed t implies that $p_{\eta _{j_1}}, p_{\eta _{j_2}}$ are linearly dependent, and additionally $p_{\eta _{j_1}}/\alpha _{\eta _{j_1}} = p_{\eta _{j_2}}/\alpha _{\eta _{j_2}}$ . Since $\alpha _{\eta _{j_1}}, \alpha _{\eta _{j_2}}$ are coprime, the good ergodicity property of $\mathopen {}( T_j^{p_j(n)} \mathclose {})_{j\in [\ell ]}$ implies that

$$ \begin{align*} {\mathcal I}(T_{\eta_{j_1}}^{\alpha_{\eta_{j_1}}} T_{\eta_{j_2}}^{-\alpha_{\eta_{j_2}}})={\mathcal I}(T_{\eta_{j_1}})\cap{\mathcal I}(T_{\eta_{j_2}}). \end{align*} $$

The equality in equation (46) follows from this and the identification in equation (47).

Example 6. (Type reduction for non-monic polynomials)

We present one more example to show how Proposition 6.1 is applied iteratively for more complicated tuples, and how properties listed in Proposition 6.4 are retained when passing to lower-type descendant tuples. Consider the tuple

(48) $$ \begin{align} (T_1^{n^2}, T_2^{3n^2}, T_3^{2n^2}, T_4^{2n^2+n}, T_5^{n^2+n}, T_6^{n^2+n}, T_7^n), \end{align} $$

and assume that it has the good ergodicity property, that is,

$$ \begin{align*} {\mathcal I}(T_1 T_2^{-3}) = {\mathcal I}(T_1)\cap{\mathcal I}(T_2),\quad {\mathcal I}(T_1 T_3^{-2}) = {\mathcal I}(T_1)\cap{\mathcal I}(T_3),\\ {\mathcal I}(T_2^3 T_3^{-2}) = {\mathcal I}(T_2)\cap{\mathcal I}(T_3),\quad {\mathcal I}(T_5 T_6^{-1}) = {\mathcal I}(T_5)\cap{\mathcal I}(T_6). \end{align*} $$

This tuple has length 7, degree 2, $K_1 = 5$ , $K_2 = 4$ , $K_3=6$ , and ${\mathfrak L} = \{1, 2, 3, 4, 5, 6\}$ . If we define the partition ${\mathfrak I}_1=\{1, 2, 3\}$ , ${\mathfrak I}_2 = \{5,6\}$ , ${\mathfrak I}_3 = \{4\}$ , ${\mathfrak I}_4 = \{7\}$ , then the tuple has type $w_0 = (3,2, 1)$ ; we recall that the term $T_7^n$ plays no part in the type consideration since the polynomial $\rho _{07}(n)=n$ has a lower degree. (Perhaps a more natural way to define the partition would be to have ${\mathfrak I}_1=\{1, 2, 3\}$ , ${\mathfrak I}_2 = \{4\}$ , ${\mathfrak I}_3 = \{5,6\}$ , ${\mathfrak I}_4 = \{7\}$ . However, then the tuple would have type $(3, 1, 2)$ , and reducing to the tuple of basic type by iteratively applying Proposition 6.1 would take more steps. This shows that choosing the partition strategically can save on the number of iterations of Proposition 6.1 needed to reach a tuple of basic type.) The tuple in equation (48) has the basic indexing tuple $\eta _0 = (1, 2, 3, 4, 5, 6,7)$ . The tuple is controllable, and 4 satisfies the controllability condition, so in the first step, we replace $T_4$ (this corresponds to us wanting to first get a $T_4$ -seminorm control over the tuple in equation (48)). We are then provided with an index $i_0\in {\mathfrak I}_1\cup {\mathfrak I}_2$ (say, $i_0 = 1$ ), and we get the new indexing tuple

$$ \begin{align*} \eta_1 := \tau_{m_0 i_0} \eta_0 = \tau_{41}\eta_0 = (1, 2, 3, 1, 5, 6, 7). \end{align*} $$

The leading coefficient 2 of $2n^2+n$ does not divide the linear coefficient, and the smallest $\unicode{x3bb} _0\in {\mathbb N}$ such that $2$ divides the coefficients of $\unicode{x3bb} _0(2n^2+n)$ is $\unicode{x3bb} _0 = 2$ . In performing the ping step of the seminorm smoothing argument for the tuple in equation (48), we will want to apply the $T_1 T_4^{-2}$ invariance of some function u to replace $T_4^{2n^2+n}u$ by $T_1^{q(n)}u'$ for some $q\in {\mathbb Z}[n]$ and a function $u'$ related in some way to u. We cannot do this directly since $\tfrac 12(2n^2+n)\notin {\mathbb Z}[n]$ , but we can do this ‘piecewise’ by splitting ${\mathbb N}$ into arithmetic progressions $(2{\mathbb N} + r)_{r=0, 1}$ and considering the two cases separately (see the sketch of the seminorm smoothing argument for $n^2, n^2, 2n^2+n$ at the end of §4 to see how this was done for that family). We therefore replace the original polynomials $\rho _{01}, \ldots , \rho _{07}$ by new polynomials

$$ \begin{align*} \rho_{1j}(n) := \begin{cases} \rho_{0j}(2 n + r_0) - \rho_{0j}(r_0), &j \neq 4,\\ \frac{1}{2}(\rho_{04}(2 n + r_0) - \rho_{04}(r_0)), &j =4, \end{cases} \end{align*} $$

for some $r_0\in \{0,1\}$ (the choice of $r_0$ is not ours). Assuming that $r_0=1$ , we obtain the new tuple

(49) $$ \begin{align} (T_1^{4n^2 + 4n}, T_2^{12n^2+12n}, T_3^{8n^2+8n}, T_1^{4n^2+5n}, T_5^{4n^2+6n}, T_6^{4n^2+6n}, T_7^{2n}). \end{align} $$

The type of the new tuple is $w_1 = \sigma _{31}w_0 = (4, 2,0)$ since we now have four transformations with indices coming from ${\mathfrak I}_1$ and two transformations coming from ${\mathfrak I}_2$ . This type is lower than the original type $w_0$ , and so we have successfully obtained a tuple of lower type. The new tuple is controllable, with $m=5,6$ both satisfying the controllability condition.

Although we replaced the polynomials $\rho _{01}, \ldots , \rho _{07}$ by new ones, we note that for any $j_1, j_2\in [7]$ , the polynomials $\rho _{1j_1}, \rho _{1j_2}$ are pairwise dependent if and only if $\rho _{0j_1}, \rho _{0j_2}$ are, and not only that: if they are pairwise dependent, then $\rho _{1j_1}/c_1 = \rho _{1j_2}/c_2$ if and only if $\rho _{0j_1}/c_1 = \rho _{0j_2}/c_2$ for any non-zero integers $c_1, c_2$ . Moreover, if $\eta _{1j_1} = \eta _{1j_2}$ , then the leading coefficients of $\rho _{1j_1}$ and $\rho _{1j_2}$ are identical. These two observations ensure that the ergodicity conditions on $T_1 T_2^{-3}, T_1 T_3^{-2}, T_2^3 T_3^{-2}, T_5 T_6^{-1}$ , which constitute the assumption that the original tuple in equation (48) has the good ergodicity property, carry on to the new tuple in equation (49), implying that it also enjoys the good ergodicity property. This exemplifies the claim from Proposition 6.4 that descendants of tuples with the good ergodicity property inherit the property.

The type $w_1$ is not basic, and so we continue the procedure. This time, we pick some $m_1\in {\mathfrak I}_2$ , say $m_1 = 5$ (it satisfies the controllability condition, as does 6, the other possible choice). We are then handed an index $i_1\in {\mathfrak I}_1$ (say, $i_1 = 3$ ), so that

$$ \begin{align*} \eta_2 := \tau_{m_1 i_1}\eta_1= \tau_{53}\eta_1 = (1, 2, 3, 1, 3, 6, 7). \end{align*} $$

The leading coefficient $4$ of $4n^2+6n$ does not divide the linear term, and so we replace the polynomials $\rho _{11}, \ldots , \rho _{17}$ by new polynomials of the form

$$ \begin{align*} \rho_{2j}(n) := \begin{cases} \rho_{1j}(2 n + r_1) - \rho_{1j}(r_1), &j \neq 5,\\ \frac{8}{4}(\rho_{15}(2 n + r_1) - \rho_{15}(r_1)), &j =5, \end{cases} \end{align*} $$

for some $r_1\in \{0,1\}$ (we pass from n to $2n+r_1$ because 2 is the smallest natural number $\unicode{x3bb} _1$ such that the leading coefficient 4 of $\rho _{15}$ divides the coefficients of $\unicode{x3bb} _1 \rho _{15}$ ). Hence, the new tuple takes the form (upon assuming $r_1 = 0$ )

(50) $$ \begin{align} (T_1^{16n^2 + 8n}, T_2^{48n^2+24n}, T_3^{32n^2+16n}, T_1^{16n^2+10n}, T_3^{32n^2+24n}, T_6^{16n^2+12n}, T_7^{4n}). \end{align} $$

The tuple in equation (50) still has the good ergodicity property; this is once again a consequence of two facts:

• • for any $j_1, j_2\in [7]$ and non-zero $c_1, c_2\in {\mathbb Z}$ , we have $\rho _{2j_1}/c_1 = \rho _{2j_2}/c_2$ if and only if $\rho _{0j_1}/c_1 = \rho _{0j_2}/c_2$ ;

• • for any $j_1, j_2\in [7]$ , if $\eta _{2j_1} = \eta _{2j_2}$ , then $\rho _{2j_1}, \rho _{2j_2}$ have identical leading coefficients.

We remark though that to ensure the ergodicity property of equation (50), we no longer need the original assumption ${\mathcal I}(T_5 T_6^{-1}) = {\mathcal I}(T_5)\cap {\mathcal I}(T_6)$ because the transformation $T_5$ is not present.

The new tuple in equation (50) has type $w_2 = \sigma _{21}w_1 = (5,1, 0)$ , which is still not basic, and so we continue the procedure one more time. The only index left in ${\mathfrak I}_2$ is $6$ , and it satisfies the controllability assumption, so we replace $T_6$ this time. We are given an index $i_2\in {\mathfrak I}_1$ (say, $i_2 = 1$ ), so that

$$ \begin{align*} \eta_3 := \tau_{m_2 i_2}\eta_1 = (1, 2, 3, 1, 3, 1, 7). \end{align*} $$

Since the leading coefficient 16 of $\rho _{21}$ does not divide the coefficients of the polynomial $\rho _{26}(n) = 16n^2 + 12 n$ , and the smallest $\unicode{x3bb} _2\in {\mathbb N}$ for which 16 divides the coefficients of $\unicode{x3bb} _2 \rho _{26}(n) = \unicode{x3bb} _2(16n^2+12n)$ is $\unicode{x3bb} _2 = 4$ , we define the new polynomials to be

$$ \begin{align*} \rho_{3j}(n) := \begin{cases} \rho_{2j}(4 n + r_2) - \rho_{2j}(r_2), &j \neq 6,\\[4pt] \dfrac{16}{16}(\rho_{26}(4 n + r_2) - \rho_{26}(r_2)), &j =6, \end{cases} \end{align*} $$

for some $r_2 \in \{0, 1, 2, 3\}$ . Assuming, say, $r_2 = 3$ , we get the new tuple

$$ \begin{align*} (T_1^{256n^2 + 416n}, T_2^{768n^2+1248n}, T_3^{512n^2+832n}, T_1^{256n^2+424n}, T_3^{512n^2+864n}, T_1^{256n^2+432n}, T_7^{16n}). \end{align*} $$

This tuple has the basic type $w_2 = \sigma _{21}w_2 = (6,0,0)$ , and so the procedure halts. A similar argument as before shows also that it enjoys the good ergodicity property.

Lastly, we observe that $\eta _3|_{{\mathfrak I}_1} = \eta _3|_{\{1, 2, 3\}}$ is constant, that is, while performing the type reduction procedure, we did not replace the transformations at indices from ${\mathfrak I}_1$ . This is a special case of property (vi) from Proposition 6.8, which will play an important role in the proof of Proposition 7.2, a seminorm control argument for tuples of basic type.

6.2 The role of invariance properties

Proposition 6.1 ensures that the lower type tuples to which we pass in the ping step of the smoothing argument have the good ergodicity property. However, this is not enough. For more complicated tuples, we also need to assume that the functions appearing in the associated average have some invariance properties, otherwise the induction breaks. The example that we present now displays the necessity of this extra information. We sketch how—reducing the original tuple to tuples of shorter length or lower type—we eventually arrive at averages for which we cannot obtain seminorm control unless the functions appearing in the averages satisfy certain invariance properties. We emphasize that our goal in this example is not to give a complete proof of seminorm control, but rather to point out the necessity of the invariance assumptions. Therefore, we assume without proof when convenient that we have seminorm control over certain tuples of lower type or shorter length.

Example 7. (The necessity of invariance properties)

Consider the average

(51) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}T_1^{n^2}f_1 \cdot T_2^{n^2}f_2 \cdot T_3^{n^2+n}f_3 \cdot T_4^{n^2 + n} f_4. \end{align} $$

It has length 4, degree 2, and type $w=(2,2)$ , corresponding to the partition ${{\mathfrak I}_1 = \{1, 2\}, {\mathfrak I}_2 = \{3, 4\}}$ . Suppose that equation (51) has the good ergodicity property, that is,

$$ \begin{align*} {\mathcal I}(T_1 T_2^{-1}) = {\mathcal I}(T_1)\cap {\mathcal I}(T_2)\quad \textrm{and}\quad {\mathcal I}(T_3 T_4^{-1}) = {\mathcal I}(T_3)\cap{\mathcal I}(T_4). \end{align*} $$

We illustrate the steps that need to be taken to show that this average is controlled by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_4|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s, T_4}$ for some $s\in {\mathbb N}$ .

By Proposition 3.7, the average from equation (51) is controlled by the seminorm $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_4|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{{\textbf {b}}_1}, \ldots , {{\textbf {b}}_{s+1}}}$ for some vectors

$$ \begin{align*} {\textbf{b}}_1, \ldots, {\textbf{b}}_{s+1}\in\{{\mathbf{e}}_4, {\mathbf{e}}_4-{\mathbf{e}}_3, {\mathbf{e}}_4-{\mathbf{e}}_2, {\mathbf{e}}_4-{\mathbf{e}}_1\}. \end{align*} $$

We want to replace the vector ${\textbf {b}}_{s+1}$ by (multiple copies of) ${\mathbf {e}}_4$ . Iterating this procedure gives a control of equation (51) by a $T_4$ -seminorm of $f_4$ .

If ${\textbf {b}}_{s+1}={\mathbf {e}}_4$ , there is nothing to check. If ${\textbf {b}}_{s+1} = {\mathbf {e}}_4-{\mathbf {e}}_3$ , then this is the consequence of the good ergodicity property of the average and Lemma 2.2. So the only cases to check are when ${\textbf {b}}_{s+1}$ equals ${\mathbf {e}}_4-{\mathbf {e}}_2$ or ${\mathbf {e}}_4-{\mathbf {e}}_1$ . Without loss of generality, we assume that ${\textbf {b}}_{s+1} = {\mathbf {e}}_4-{\mathbf {e}}_2$ .

Suppose that the limit of equation (51) is non-zero. Arguing as in the proof of Proposition 4.1, we deduce that

$$ \begin{align*} \liminf_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}},{\underline{h}}'\in [H]^{s}}\lim_{N\to\infty}\Big\Vert \mathop{\mathbb{E}}\limits_{n\in[N]} \, T_1^{n^2} f_{1, {\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2} f_{2, {\underline{h}}, {\underline{h}}'}\cdot T_3^{n^2+n}f_{3, {\underline{h}}, {\underline{h}}'}\cdot T_4^{n^2+n}u_{{\underline{h}},{\underline{h}}'}\Big\Vert_{L^2(\mu)}\kern1.2pt{>}\kern1.2pt0 \end{align*} $$

for some $T_4 T_2^{-1}$ -invariant functions $u_{{\underline {h}}, {\underline {h}}'}$ as well as functions $f_{j, {\underline {h}},{\underline {h}}'} := \Delta _{{{\textbf {b}}_1}, \ldots , {{\textbf {b}}_{s}}; {\underline {h}}-{\underline {h}}'} f_j$ for $j\in [4]$ . The invariance property of the functions $u_{{\underline {h}}, {\underline {h}}'}$ implies that

(52) $$ \begin{align} \liminf_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}},{\underline{h}}'\in [H]^{s}}\lim_{N\to\infty}\Big\Vert\! \mathop{\mathbb{E}}\limits_{n\in[N]} T_1^{n^2} f_{1, {\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2} f_{2, {\underline{h}}, {\underline{h}}'}\cdot T_3^{n^2+n}f_{3, {\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2+n}u_{{\underline{h}},{\underline{h}}'}\Big\Vert_{L^2(\mu)}>0. \end{align} $$

Each of the averages inside the liminf above is of the form

(53) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}T_1^{n^2} g_1\cdot T_2^{n^2} g_2\cdot T_3^{n^2+n}g_3\cdot T_2^{n^2+n}g_4 \end{align} $$

for 1-bounded functions $g_1, g_2, g_3, g_4\in L^{\infty }(\mu )$ of which $g_4$ is $T_4 T_2^{-1}$ invariant. The averages from equation (53) are controllable, with $3$ satisfying the controllability condition, and they have type $(3, 1)$ , which is lower than the type $(2,2)$ of the original average from equation (51). Assuming inductively that we have the seminorm control of averages from equation (53) by a $T_3$ -seminorm of $f_3$ (while we only use this particular control, our inductive assumption will guarantee that we control averages from equation (53) by a relevant seminorm of other functions, too), we can deduce from equation (52) (like in the proof of Proposition 4.2) that

$$ \begin{align*} \lvert\hspace{-1.2pt}|\hspace{-1.2pt}| f_3|\hspace{-1.2pt}|\hspace{-1.2pt}\rvert_{{\textbf{b}}_1, \ldots, {\textbf{b}}_s, {\mathbf{e}}_3^{\times s_1}}>0 \end{align*} $$

for some $s_1\in {\mathbb N}$ , and similarly for other terms. This completes the ping step. For the pong step, this auxiliary control and Proposition 3.3 imply that

(54) $$ \begin{align} &\liminf_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}},{\underline{h}}'\in [H]^{s}}\lim_{N\to\infty}\bigg\Vert \mathop{\mathbb{E}}\limits_{n\in[N]} \, T_1^{n^2} f_{1, {\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2} f_{2, {\underline{h}}, {\underline{h}}'}\nonumber\\ &\quad\times \prod_{j=1}^{2^{s}}{\mathcal D}_{j}(n^2+n)\cdot T_4^{n^2+n}f_{4, {\underline{h}}, {\underline{h}}'}\bigg\Vert_{L^2(\mu)}>0 \end{align} $$

for some ${\mathcal D}_j\in {\mathfrak D}_{s_1}$ . Each average in equation (54) takes the form

(55) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]} \, T_1^{n^2} g_1\cdot T_2^{n^2} g_2 \cdot \prod_{j=1}^{2^{s}}{\mathcal D}_{j}(n^2+n)\cdot T_4^{n^2+n} g_4. \end{align} $$

Assuming inductively that averages of the form in equation (55) are controlled by a $T_4$ -seminorm of the last term, we get the desired claim $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| f_4|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\textbf {b}}_1, \ldots , {\textbf {b}}_s, {\mathbf {e}}_4^{\times s'}}>0$ for some $s'\in {\mathbb N}$ using an argument similar to one in the proof of Proposition 4.2.

We have showed how a seminorm control of the original average from equation (51) by a $T_4$ -seminorm of $f_4$ follows from the seminorm control of the averages from equations (53) and (55). We have not proved, however, that these auxiliary averages are indeed controlled by Gowers–Host–Kra seminorms, assuming instead that this follows by induction. It turns out that obtaining a seminorm control of the averages from equation (53) involves an interesting twist in that the argument makes essential use of the assumption that the function $g_4$ is $T_4 T_2^{-1}$ -invariant. We sketch the steps taken in the seminorm smoothing argument for this average under the extra invariance assumption to show where this invariance property comes up and why it is necessary.

In proving the seminorm control of equation (53), we first prove that the average is controlled by a $T_3$ -seminorm of $g_3$ since $T_3$ is the only transformation with index in ${\mathfrak I}_2$ . Arguing as above (using Proposition 3.7 for equation (53), assuming that the $L^2(\mu )$ limit of equation (53) is non-zero and mimicking the proof of Proposition 4.2), we deduce that

$$ \begin{align*} \liminf_{H\to\infty}\mathop{\mathbb{E}}\limits_{{\underline{h}},{\underline{h}}'\in [H]^{s}}\lim_{N\to\infty}\Big\Vert\! \mathop{\mathbb{E}}\limits_{n\in[N]} T_1^{n^2} g_{1, {\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2} g_{2, {\underline{h}}, {\underline{h}}'}\cdot T_3^{n^2+n}u_{{\underline{h}}, {\underline{h}}'}\cdot T_2^{n^2+n}g_{4, {\underline{h}},{\underline{h}}'}\Big\Vert_{L^2(\mu)}>0 \end{align*} $$

for 1-bounded functions $u_{{\underline {h}}, {\underline {h}}'}$ that are all invariant either under $T_3 T_1^{-1}$ or under $T_3 T_2^{-1}$ . Then we use the relevant invariance property to replace each $T_3^{n^2+n} u_{{\underline {h}}, {\underline {h}}'}$ by $T_1^{n^2+n} u_{{\underline {h}}, {\underline {h}}'}$ or $T_2^{n^2+n} u_{{\underline {h}}, {\underline {h}}'}$ . Hence, in the ping step of the seminorm smoothing argument for equation (53), we need to invoke seminorm control of averages of the form

(56) $$ \begin{align} \nonumber &\mathop{\mathbb{E}}\limits_{n\in[N]}T_1^{n^2} g_1'\cdot T_2^{n^2} g_2'\cdot T_1^{n^2+n}g_3'\cdot T_2^{n^2+n}g_4'\\ \textrm{and}\quad & \mathop{\mathbb{E}}\limits_{n\in[N]}T_1^{n^2} g_1'\cdot T_2^{n^2} g_2'\cdot T_2^{n^2+n}g_3'\cdot T_2^{n^2+n}g_4', \end{align} $$

where $g^{\prime }_4$ is $T_4 T_2^{-1}$ -invariant while $g^{\prime }_3$ is invariant under $T_3 T_1^{-1}$ and $T_3 T_2^{-1}$ respectively. Both of them have basic type.

We show that for arbitrary $g_1', g_2', g_3', g_4'$ , without the aforementioned invariance assumptions, we would not be able to control the average from equation (56) by Gowers–Host–Kra seminorms; specifically, we could not control it by a $T_2$ -seminorm of $g^{\prime }_4$ . Conversely, this is achievable if $g^{\prime }_3, g^{\prime }_4$ are invariant under $T_3T_2^{-1}$ , $T_4T_2^{-1}$ , respectively. Assuming for simplicity that $g_1'=g_2':= 1$ , we have that the second average equals

$$ \begin{align*} \mathop{\mathbb{E}}\limits_{n\in[N]}T_1^{n^2} g_1'\cdot T_2^{n^2} g_2'\cdot T_2^{n^2+n}g_3'\cdot T_2^{n^2+n}g_4' = \mathop{\mathbb{E}}\limits_{n\in[N]}T_2^{n^2+n}(g_3'\cdot g_4'), \end{align*} $$

and so without any additional assumptions, the average from equation (56) is in general not controlled by a $T_2$ -seminorm of $g_3'$ or a $T_2$ -seminorm of $g^{\prime }_4$ . However, the invariance assumptions on $g^{\prime }_3, g^{\prime }_4$ give us

(57) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}T_2^{n^2+n}(g_3'\cdot g_4') = \mathop{\mathbb{E}}\limits_{n\in[N]} T_3^{n^2+n}g_3'\cdot T_4^{n^2+n}g_4', \end{align} $$

and by Proposition 3.7, these averages can be controlled by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\mathbf {e}}_4^{\times s}, ({\mathbf {e}}_4-{\mathbf {e}}_3)^{\times s}}$ for some $s\in {\mathbb N}$ . Then the assumption ${\mathcal I}(T_4 T_3^{-1})\subseteq {\mathcal I}(T_4)$ and Lemma 2.2 give $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{{\mathbf {e}}_4^{\times s}, ({\mathbf {e}}_4-{\mathbf {e}}_3)^{\times s}} \leq \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g^{\prime }_4|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{2s, T_4}$ , and so a $T_4$ -seminorm of $g_4'$ does control the average from equation (57). Using the invariance property once again, this time along with Lemma 3.5, we get $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_4} =\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_2}$ , so a $T_2$ -seminorm of $g^{\prime }_4$ controls equation (57) and hence equation (56).

To get control over equation (56) by a $T_2$ -seminorm of $g_4'$ without any simplifying assumptions on $g_1', g_2'$ , we have to run a more complicated argument. Combining Proposition 3.7, the ergodic condition on $T_1T_2^{-1}$ and Lemma 2.2, we first obtain control of equation (56) by a $T_1$ -seminorm of $g^{\prime }_1$ and a $T_2$ -seminorm of $g^{\prime }_2$ . Assuming that the $L^2(\mu )$ limit of the average from equation (56) is positive, we use this newly established control, decompose $g^{\prime }_1$ using Proposition 2.3, and apply the pigeonhole principle to show that the average

(58) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}{\mathcal D}(n^2)\cdot T_2^{n^2} g_2'\cdot T_2^{n^2+n}g_3'\cdot T_2^{n^2+n}g_4' \end{align} $$

has a non-vanishing limit. The invariance properties of $g^{\prime }_3$ and $g^{\prime }_4$ imply that the average from equation (58) equals

(59) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}{\mathcal D}(n^2)\cdot T_2^{n^2} g_2'\cdot T_3^{n^2+n}g_3'\cdot T_4^{n^2+n}g_4', \end{align} $$

for which a seminorm control by a $T_4$ -seminorm of $g^{\prime }_4$ follows by inductively invoking seminorm control for averages of length 3. The invariance property of $g^{\prime }_4$ implies once again that $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_4} =\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4'|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_2}$ for any $s'\in {\mathbb N}$ . It follows that a $T_2$ -seminorm of $g^{\prime }_4$ controls equation (59), and hence also equations (58) and (56).

The invariance properties also come up in the pong step of the smoothing argument for equation (53). In this part, we encounter averages of the form

(60) $$ \begin{align} \mathop{\mathbb{E}}\limits_{n\in[N]}\prod_{j=1}^L {\mathcal D}_{j}(n^2)\cdot T_2^{n^2} g_2"\cdot T_3^{n^2+n}g_3"\cdot T_2^{n^2+n}g_4". \end{align} $$

Moreover, the function $g_4"$ is $T_4 T_2^{-1}$ -invariant because it is essentially a multiplicative derivative of $g_4'$ . By a similar reason as before, such averages could not be controlled by Gowers–Host–Kra seminorms for arbitrary $g_2", g_3", g_4"$ without the invariance assumption. However, thanks to the invariance assumption, the average from equation (60) equals

(61) $$ \begin{align} \lim_{N\to\infty} \mathop{\mathbb{E}}\limits_{n\in[N]}\prod_{j=1}^L {\mathcal D}_{j}(n^2)\cdot T_2^{n^2} g_2"\cdot T_3^{n^2+n}g_3"\cdot T_4^{n^2+n}g_4", \end{align} $$

which is controlled by $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g^{\prime \prime }_4|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_4}$ for some $s'\in {\mathbb N}$ . (We can assume this inductively, or we can prove that a $T_4$ -seminorm of $g_4"$ controls equation (61) in essentially the same way as we argued in Proposition 4.2.) Using the invariance property of $g^{\prime \prime }_4$ again together with Lemma 3.5, we deduce that $\lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4"|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_4} = \lvert \hspace {-1.2pt}|\hspace {-1.2pt}| g_4"|\hspace {-1.2pt}|\hspace {-1.2pt}\rvert _{s', T_2}$ , and so a $T_2$ -seminorm of $g^{\prime \prime }_4$ does control equation (60). An argument similar to the one at the end of the proof of Proposition 4.2 implies that a $T_2$ -seminorm of $g_4$ controls equation (53).