Proof of Theorem 2.4 assuming Theorem 3.1

Let $a_1,\ldots , a_\ell $ and $w_N(n)$ be as in Theorem 2.4. Since the assumptions of Theorem 2.4 are symmetric with respect to the sequences $a_1,\ldots , a_\ell $ , it suffices to show that there exists $s\in {\mathbb N}$ such that if $\lvert \!|\!| f_1|\!|\!\rvert _s=0$ , then equation (7) holds.

If $a_1$ has maximal fractional degree within the family $a_1,\ldots a_\ell $ , then if we take $k=0$ and all functions to be independent of N in Theorem 3.1, we get that the conclusion of Theorem 2.4 holds. Otherwise, we can assume that $a_{\ell }$ is the function with the highest fractional degree and, as a consequence, $\text {f-deg}(a_1)<\text {f-deg}(a_\ell )$ . It suffices to show that

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{n\in [N]}\, w_{N}(n)\cdot \int f_{N,0} \cdot \prod_{i=1}^\ell T^{[a_i(n)]}f_{i} \, d\mu=0, \end{align*} $$

where

$$ \begin{align*} f_{N,0} := {\mathbb E}_{n\in [N]} \, \overline{w}_{N}(n)\cdot \prod_{i=1}^\ell T^{[a_i(n)]}\overline{f}_i, \quad N\in {\mathbb N}. \end{align*} $$

Note that since $f_1,\ldots , f_\ell $ and $c_N$ are $1$ -bounded, we have

$$ \begin{align*} \limsup_{N\to\infty} \left\Vert f_{N,0}\right\Vert_\infty\leq \lim_{N\to\infty} {\mathbb E}_{n\in [N]} \, \Lambda'(n)=1, \end{align*} $$

(the last identity follows from the prime number theorem, but we only need the much simpler upper bound) hence, we can assume that $f_{N,0}$ is $1$ -bounded for every $N\in {\mathbb N}$ .

After composing with $T^{-[a_{\ell }(n)]}$ , using the Cauchy-Schwarz inequality, and the identity $[x]-[y]=[x-y]+e$ for some $e\in \{0,1\}$ , we are reduced to showing that

$$ \begin{align*} \lim_{N\to\infty} \left\Vert {\mathbb E}_{n\in [N]}\, w_{N}(n)\cdot \prod_{i=1}^{\ell-1} T^{[a_i(n)-a_\ell(n)]+e_i(n)}f_i\cdot T^{[-a_{\ell}(n)]+e_\ell(n)}f_{N,0}\right\Vert_{L^2(\mu)}=0, \end{align*} $$

for some $e_1(n),\ldots , e_{\ell -1}(n)\in \{0,1\}$ , $n\in {\mathbb N}$ . Next, we would like to replace the error sequences $e_1,\ldots , e_{\ell -1}$ with constant sequences. To this end, we use Lemma 3.6 for I a singleton, $J:=[N]$ , $X:=L^\infty (\mu )$ , $A_N(n_1,\ldots ,n_\ell ):=\prod _{i=1}^{\ell -1} T^{n_i}f_i\cdot T^{-n_\ell }f_{N,0}$ , $n_1,\ldots , n_\ell \in {\mathbb Z}$ , and $b_i:=[a_i-a_{\ell }]$ , $i=1,\ldots , \ell -1$ , $b_\ell := [-a_\ell ]$ . We get that it suffices to show that

(9) $$ \begin{align} \lim_{N\to\infty} \left\Vert {\mathbb E}_{n\in [N]}\, z_{N}(n)\cdot \prod_{i=1}^\ell T^{[a^{\prime}_i(n)]}g_{N,i}\right\Vert_{L^2(\mu)}=0, \end{align} $$

where

$$ \begin{align*} a^{\prime}_i:=a_i-a_{\ell}, \quad i=1,\ldots, \ell-1, \quad a^{\prime}_\ell:= -a_\ell \end{align*} $$

for some $1$ -bounded sequence $(z_{N}(n))$ , where $g_{N,i}:=T^{\epsilon _i}f_{i}$ , $i=1,\ldots , \ell -1$ , $g_{N,\ell }:=T^{\epsilon _\ell }f_{N,0}$ , $N\in {\mathbb N}$ , for some constants $\epsilon _1,\ldots , \epsilon _\ell \in \{0,1\}$ . Note that the family $a^{\prime }_1,\ldots , a^{\prime }_\ell $ is nice, and $g_{N,1}=T^{\epsilon _1}f_1$ , $N\in {\mathbb N}$ , so Theorem 3.1 applies (for $k=0$ and all but one of the functions independent of N) and gives that there exists $s\in {\mathbb N}$ so that if $\lvert \!|\!| f_1|\!|\!\rvert _s=0$ , then equation (9) holds. This completes the proof.

Proof. In the following argument, whenever we write p, we assume that p is a prime number.

Let ${\underline {h}}\in [N]^\ell $ . Note that if $\nu _p(\text {cube}({\underline {h}}))=2^\ell $ , then

$$ \begin{align*} \Big(1-\frac{1}{p}\Big)^{-2^\ell}\Big(1-\frac{\nu_p(\text{cube}({\underline{h}}))}{p}\Big)\leq 1, \end{align*} $$

and if $\nu _p(\text {cube}({\underline {h}}))<2^\ell $ , then for $a_\ell :=2^{\ell +1}-2$ , we have

$$ \begin{align*} \Big(1-\frac{1}{p}\Big)^{-2^\ell}\Big(1-\frac{\nu_p(\text{cube}({\underline{h}}))}{p}\Big)\leq \Big(1-\frac{1}{p}\Big)^{-(2^\ell-1)}\leq e^{\frac{a_\ell}{p}}, \end{align*} $$

where we use that $\frac {1}{1-x}\leq e^{2x}$ for $x\in [0,\frac {1}{2}]$ . Note also that if $\nu _p(\text {cube}({\underline {h}}))<2^\ell $ , then there exist distinct $\underline \epsilon ,\underline \epsilon '\in \{0,1\}^{\ell }$ such that $p|(\underline \epsilon -\underline \epsilon ')\cdot {\underline {h}}$ , in which case we have that $p\in \mathcal {P}({\underline {h}})$ , where

$$ \begin{align*} \mathcal{P}({\underline{h}}):=\bigcup_{\underline\epsilon,\underline\epsilon' \in \{0,1\}^\ell, \underline\epsilon,\neq \underline\epsilon' }\, \{p\in {\mathbb P} \colon p|(\underline\epsilon-\underline\epsilon')\cdot {\underline{h}}\}, \quad {\underline{h}}\in {\mathbb N}^\ell. \end{align*} $$

We deduce from the above facts and equation (10) that

(12) $$ \begin{align} \mathfrak{G}_{2^\ell}(\text{cube}({\underline{h}})) \leq e^{a_\ell \sum_{p\in \mathcal{P}({\underline{h}})}\frac{1}{p}}. \end{align} $$

By [Reference Tao and Ziegler27, Lemma E.1], we have for some $b_\ell ,c_\ell>0$ that

(13) $$ \begin{align} e^{a_\ell \sum_{p\in \mathcal{P}({\underline{h}})}\frac{1}{p}} \leq b_\ell \sum_{p\in \mathcal{P}({\underline{h}})}\frac{(\log{p})^{c_\ell}}{p}= b_\ell \sum_{\underline\epsilon,\underline\epsilon' \in \{0,1\}^\ell, \underline\epsilon\neq \underline\epsilon'}\Big(\sum_{ p|(\underline\epsilon-\underline\epsilon')\cdot {\underline{h}}}\frac{(\log{p})^{c_\ell}}{p}\Big). \end{align} $$

Moreover, we get for some $d_\ell ,e_\ell>0$ that

(14) $$ \begin{align} \sum_{{\underline{h}}\in [N]^\ell}\Big(\sum_{ p|(\underline\epsilon-\underline\epsilon')\cdot {\underline{h}}}\frac{(\log{p})^{c_\ell}}{p}\Big)\leq d_\ell \sum_{ p}\frac{(\log{p})^{c_\ell}}{p} \frac{N^\ell}{p}\leq e_\ell\, N^\ell, \end{align} $$

for all $N\in {\mathbb N}$ , where to get the first estimate, we used the fact that for some $d_\ell>0$ , we have

$$ \begin{align*} |{\underline{h}}\in[N]^\ell\colon p|(\underline\epsilon-\underline\epsilon')\cdot {\underline{h}}|\leq d_\ell \frac{N^\ell}{p} \end{align*} $$

for all $N\in {\mathbb N}$ , and to get the second estimate, we used that $\sum _{ p}\frac {(\log {p})^{c_\ell }}{p^2}<\infty $ .

If we take squares in equation (12), sum over all ${\underline {h}}\in [N]^\ell $ and then use equations (13) and (14), we get the asserted estimate.

Proof. Since $\Lambda '$ is supported on primes and $c+h_1+\cdots +h_\ell \leq N^A$ , we have that

$$ \begin{align*} \sum_{n\in [N]}\, (\Delta_{\underline{h}}\Lambda')(n+c)\leq |\{n\in [N]\colon \underline{n+c}+\text{cube}({\underline{h}})\in {\mathbb P}^{2^\ell}\}|\cdot (\log(N+N^A))^{2^\ell}, \end{align*} $$

where $\underline {n+c}$ is a vector with $2^\ell $ coordinates, all equal to $n+c$ . Note that for ${\underline {h}}\in ({\mathbb N}^\ell )^*$ , we can apply Theorem 3.2, and we get that there exists $D_{A,\ell }>0$ such that for every $N\in {\mathbb N}$ , the last expression is bounded by

$$ \begin{align*} D_{A,\ell}\, \mathfrak{G}_{2^\ell}(\text{cube}({\underline{h}}))\, N. \end{align*} $$

If we let $C_{A,\ell }({\underline {h}}):=D_{A,\ell }\, \mathfrak {G}_{2^\ell }(\text {cube}({\underline {h}}))$ and ${\underline {h}}\in {\mathbb N}^\ell $ and use Proposition 3.3, we get that properties $(i)$ and $(ii)$ hold.

Proof. The expression on the left-hand side is bounded by

$$ \begin{align*} \sum_{j=1}^t\sum_{{\underline{h}}\in I} \Big|\Big|\sum_{n\in J}\, w_{\underline{h}}(n)\cdot A_{\underline{h}}(b_{1,{\underline{h}}}(n)+e_{1,{\underline{h}}}(n),\ldots, b_{\ell,{\underline{h}}}(n)+e_{\ell,{\underline{h}}}(n)) \cdot \mathbf{1}_{E_{j,{\underline{h}}}} (n)\Big|\Big|, \end{align*} $$

where for $t=f^\ell $ , the sets $E_{1,{\underline {h}}},\ldots , E_{t,{\underline {h}}}$ form a partition of ${\mathbb N}$ into sets (possibly empty) on which all the sequences $e_{1,{\underline {h}}},\ldots , e_{\ell ,{\underline {h}}}$ are constant (and the constants do not depend on ${\underline {h}}$ ). If the maximum of the summands over j occurs for some $j_0\in [t]$ , then there exist $\epsilon _1,\ldots , \epsilon _\ell \in F$ such that for all $n\in E_{j_0, {\underline {h}}}$ , we have $e_{i,{\underline {h}}}(n)=\epsilon _i$ , $i \in [\ell ]$ , ${\underline {h}} \in I$ . Hence, the last sum is bounded by

$$ \begin{align*} t \sum_{{\underline{h}}\in I}\Big|\Big|\sum_{n\in J}\, \tilde{w}_{\underline{h}}(n)\cdot A_{\underline{h}}(b_{1,{\underline{h}}}(n)+\epsilon_1,\ldots, b_{\ell,{\underline{h}}}(n)+\epsilon_\ell) \Big|\Big|, \end{align*} $$

where $\tilde {w}_{\underline {h}}(n):=w_{\underline {h}}(n)\cdot \mathbf {1}_{E_{j_0,{\underline {h}}}}(n)$ , $n\in J$ , ${\underline {h}}\in I$ .

Proof. We cover the case where $w_{N,{\underline {h}}}(n)=(\Delta _{\underline {h}}\Lambda ')(n)\cdot c_{N,{\underline {h}}}(n)$ ; the case where $w_{N,{\underline {h}}}(n)= c_{N,{\underline {h}}}(n)$ is similar (in fact, easier).

By assumption, we have that $a_i({\underline {h}},t):=\sum _{j=0}^r p_{i,j}({\underline {h}})t^{d_j}$ , $i=1,\ldots , \ell $ , where $0=d_0< d_1<\ldots <d_r=d< 1$ and $p_{i,j}\in {\mathbb R}[t_1,\ldots , t_k]$ with $p_{1,r}\neq 0$ . We let

$$ \begin{align*} p({\underline{h}}):=\big[\max_{i,j}\{|p_{i,j}|({\underline{h}})\}^{\frac{1}{d_1}}\big]+1, \quad {\underline{h}}\in {\mathbb N}^k.\\[-16pt] \end{align*} $$

For ${\underline {h}}\in [L_N]^k$ , after partitioning $[N^a]$ into subintervals, we deduce that it suffices to get an upper bound for the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in I_{N,{\underline{h}}}}\, {\mathbb E}^*_{n_1\in J_{n,{\underline{h}}}}\, w_{N,{\underline{h}}}(n_1) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},n_1)]} f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)},\\[-16pt] \end{align*} $$

where

$$ \begin{align*} I_{N,{\underline{h}}}:= [N^{ad}p({\underline{h}})], \quad J_{n,{\underline{h}}}:=\Big[\Big(\frac{n-1}{p({\underline{h}})}\Big)^{\frac{1}{d}}, \Big(\frac{n}{p({\underline{h}})}\Big)^{\frac{1}{d}}\Big), \quad n\in I_{N,{\underline{h}}},\, {\underline{h}}\in [L_N]^k,\, N\in{\mathbb N},\\[-16pt] \end{align*} $$

and for $D\colon {\mathbb N}\to {\mathbb C}$ and fixed $N\in {\mathbb N}$ , $n\in I_{N,{\underline {h}}},\, {\underline {h}}\in [L_N]^k$ , we let

(19) $$ \begin{align} {\mathbb E}^*_{n_1\in J_{n,{\underline{h}}}}D(n_1) :=\frac{1}{N^a/|I_{N,{\underline{h}}}|}\sum_{n_1\in J_{n,{\underline{h}}}}D(n_1).\\[-16pt]\nonumber \end{align} $$

Note that an application of the mean value theorem gives

(20) $$ \begin{align} |J_{n,{\underline{h}}}|\leq\frac{1}{d}\frac{(N^{ad}p(h))^{\frac{1}{d}-1} }{p(h)^{\frac{1}{d}}}=\frac{1}{d} \cdot \frac{N^a}{|I_{N,{\underline{h}}}|}, \quad n\in I_{N,{\underline{h}}},\, {\underline{h}}\in [L_N]^k, \, N\in {\mathbb N}.\\[-16pt]\nonumber \end{align} $$

ForFootnote ⁷ convenience, we write

$$ \begin{align*} J_{n,{\underline{h}}}=(k_{n,{\underline{h}}},k_{n,{\underline{h}}}+l_{n,{\underline{h}}} ], \quad n\in I_{N,{\underline{h}}},\, h\in [L_N]^k,\, N\in {\mathbb N}\\[-16pt] \end{align*} $$

for some $k_{n,{\underline {h}}}, l_{n,{\underline {h}}}\in {\mathbb N}$ . Note that for $i=1,\ldots ,\ell $ , $j=1,\ldots , r$ and fixed $n,{\underline {h}}$ , when $n_1$ ranges on $J_{n,{\underline {h}}}$ , the values of $p_{i,j}({\underline {h}})n_1^{d_j}$ belong to an interval of length $1$ . Hence, for $i=1,\ldots , \ell $ , we can write

$$ \begin{align*} a_i({\underline{h}},n_1)=a_i({\underline{h}},(n/p({\underline{h}}))^{1/d})+\epsilon_i({\underline{h}},n, n_1),\\[-16pt] \end{align*} $$

where $\epsilon _i({\underline {h}},n,n_1)$ is bounded by r for all $n_1\in J_{n,{\underline {h}}}$ , $n\in I_{N,{\underline {h}}}$ , ${\underline {h}}\in [L_N]^k$ , $N\in {\mathbb N}$ .

The terms $\epsilon _i({\underline {h}},n,n_1)$ can be easily taken care by using Lemma 3.6 and appropriately modifying $(c_{N,{\underline {h}}}(n))$ to another bounded sequence of weights. We deduce that it suffices to get an upper bound for the averages

(21) $$ \begin{align} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in I_{N,{\underline{h}}}}\, \mathbf{1}_{I^{\prime}_{N,{\underline{h}}}}(n) \, \tilde{w}_{N,{\underline{h}}}(n) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},(n/p({\underline{h}}))^{1/d})]+\epsilon_{i,N}} f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)},\\[-16pt]\nonumber \end{align} $$

where $I^{\prime }_{N,{\underline {h}}}:= [N^{\frac {ad}{2}},N^{ad}p({\underline {h}})]$ , $N\in {\mathbb N}$ (the indicator introduces a negligible $o_N(1)$ term), $\epsilon _{1,N},\ldots , \epsilon _{\ell ,N}$ take finitely many values for $N\in {\mathbb N}$ , and for $n\in [I_{N,{\underline {h}}}], {\underline {h}}\in [L_N]^k, N\in {\mathbb N}$ , we let

(22) $$ \begin{align} \tilde{w}_{N,{\underline{h}}}(n):={\mathbb E}^*_{n_1\in J_{n,{\underline{h}}}}\, w_{N,{\underline{h}}}(n_1).\\[-16pt]\nonumber \end{align} $$

We used that $L_N,\Lambda '(N)\prec N^\varepsilon $ for all $\varepsilon>0$ to justify that inserting the indicator $\mathbf { 1}_{I^{\prime }_{N,{\underline {h}}}}$ only introduces an $o_N(1)$ term, which is fine for our purposes.

Using that $(c_{N,{\underline {h}}}(n))$ is $1$ -bounded, $((\Delta _{{\underline {h}}}\Lambda ')(n))$ is nonnegative and equations (19) and (20), we deduce that

(23) $$ \begin{align} |\tilde{w}_{N,{\underline{h}}}(n)|\leq d^{-1}\cdot {\mathbb E}_{n_1\in J_{n, {\underline{h}}}}\, (\Delta_{{\underline{h}}}\Lambda')(n_1)=d^{-1}\cdot {\mathbb E}_{n_1\in [l_{n,{\underline{h}}}]}\, (\Delta_{{\underline{h}}}\Lambda')(n_1+k_{n,{\underline{h}}}). \end{align} $$

From the definition of $l_{n,{\underline {h}}}$ and the mean value theorem, we have that

$$ \begin{align*} l_{n,{\underline{h}}}\geq \frac{n^{\frac{1}{d}-1}}{d(p({\underline{h}}))^{\frac{1}{d}}}, \quad n\in{\mathbb N}. \end{align*} $$

Since $L_N\prec N^\varepsilon $ for every $\varepsilon>0$ and $k_{n,{\underline {h}}}\leq n^{1/d}$ , it follows that if $A>\frac {1}{1-d}$ – for example, if $A:=\frac {1}{1-d}+1$ – then there exists $N_0=N_0(d,p)\in {\mathbb N}$ such that for all $N\geq N_0$ and all $n\in I^{\prime }_{N,{\underline {h}}}$ , ${\underline {h}}\in [L_N]^k$ , we have

$$ \begin{align*} k_{n,{\underline{h}}}\leq l_{n,{\underline{h}}}^A. \end{align*} $$

Hence,Footnote ⁸ there exists $N_1=N_1(d,k,p)\in {\mathbb N}$ such that for all $N\geq N_1$ , we have for all $n\in I^{\prime }_{N,{\underline {h}}}$ and ${\underline {h}}=(h_1,\ldots , h_k)\in [L_N]^k$ that

$$ \begin{align*} k_{n,{\underline{h}}}+h_1+\cdots + h_k \leq l_{n,{\underline{h}}}^A. \end{align*} $$

We will combine this with the identity

$$ \begin{align*} (\Delta_{{\underline{h}}}\Lambda')(n)=\prod_{\epsilon\in \{0,1\}^k}\Lambda'(n+\epsilon\cdot {\underline{h}}), \end{align*} $$

the estimate equation (23) and Corollary 3.4 (with $\ell :=k$ , $c:=k_{n,{\underline {h}}}$ , $N:=l_{n,{\underline {h}}}$ , $A:=\frac {1}{1-d}+1$ ). We deduce that there exist $C=C(d,k)>0$ and $C_{d,k}({\underline {h}})>0$ , ${\underline {h}}\in {\mathbb N}^k$ , such that for all large enough N (depending only on $d,k,p$ ), for every $n\in I_{N,{\underline {h}}}$ , ${\underline {h}}\in ([L_N]^k)^*$ , we can write

(24) $$ \begin{align} \tilde{w}_{N,{\underline{h}}}(n)=C_{d,k}({\underline{h}})\cdot z_{N,{\underline{h}}}(n), \end{align} $$

where $(z_{N,{\underline {h}}}(n))$ is $1$ -bounded and

(25) $$ \begin{align} {\mathbb E}_{{\underline{h}}\in [L_N]^k}(C_{d,k}({\underline{h}}))^2\leq C \end{align} $$

for every $N\in {\mathbb N}$ .

Note that since $L_N\succ (\log {N})^K$ for every $K>0$ and $\Lambda '(n)\leq \log {n}$ , for every $n\in {\mathbb N}$ , we have that $\max _{{\underline {h}}\in [L_N]^k, n\in [N]}(\tilde {w}_{N,{\underline {h}}}(n))^2\prec L_N$ . Using this, and since by equation (11), we have that $\frac {1}{L_N^k}|[L_N]^k\setminus ([L_N]^k)^*|\ll _k \frac {1}{L_N}$ , we deduce that we can redefine $C({\underline {h}})$ on the complement of $([L_N]^k)^*$ so that for all large enough N (depending on $d,k,p$ ), equation (24) holds for all $n\in I_{N,{\underline {h}}}$ , ${\underline {h}}\in [L_N]^k$ , and equation (25) also holds (for some larger constant $C'$ in place of C).

We now use equations (24) and (25) and the Cauchy-Schwarz inequality to bound the averages in equation (21). We can also remove the indicator $ \mathbf {1}_{I^{\prime }_{N,{\underline {h}}}}(n)$ since it has a negligible effect on our averages. We deduce that it suffices to get an upper bound for the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in I_{N,{\underline{h}}}}\, \, z_{N,{\underline{h}}}(n) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},(n/p({\underline{h}}))^{1/d})]+\epsilon_{i,N}} f_{N,{\underline{h}},i}\right\Vert^2_{L^2(\mu)}. \end{align*} $$

Note that since the weights and functions are bounded, it suffices to get an upper bound for the previous expression, ignoring the square. For ${\underline {h}}\in [L_N]^k$ , we can express $n\in I_{N,{\underline {h}}}$ as $n=n'p({\underline {h}})+s$ for some $n'\in [N^{ad}]$ or $n'=0$ and $s\in [p({\underline {h}})]$ . After renaming $n'$ as n for convenience, we are led to upper-bounding the averages

(26) $$ \begin{align} {\mathbb E}_{{\underline{h}}\in [L_N]^k}{\mathbb E}_{s\in [p({\underline{h}})]} \left\Vert {\mathbb E}_{n\in [N^{ad}]}\, \, z_{N,{\underline{h}},s}(n) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},(n+s/p({\underline{h}}))^{1/d})]+\epsilon_{i,N}} f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)} \end{align} $$

for some $1$ -bounded sequence $(z_{N,{\underline {h}},s}(n))$ . Note that if $u\in (0,1)$ and $q\in {\mathbb R}[t_1,\ldots , t_k]$ , then an application of the mean value theorem shows that for every $\varepsilon>0$ , we have

$$ \begin{align*} \lim_{N\to\infty} \sup_{c\in [0,1], {\underline{h}}\in[L_N]^k,n\geq N^{\varepsilon}} | q({\underline{h}}) ((n+c)^u-n^u) | = 0. \end{align*} $$

It follows that in equation (26), when computing $a_i({\underline {h}},(n+s/p({\underline {h}}))^{1/d})$ , we can replace $n+s/p({\underline {h}})$ with n in the nonlinear monomials; this will lead to some error sequences that are $1$ -bounded for large enough N and can be handled by appealing to Lemma 3.6 (and redefining the sequence $z_{N,{\underline {h}}}(n)$ ). With this in mind, it follows that in equation (26), we can replace $a_i({\underline {h}},(n+s/p({\underline {h}}))^{1/d})$ with $a_i({\underline {h}},n^{1/d})+\frac {p_{i,r}({\underline {h}})}{p({\underline {h}})}s$ . Hence, it suffices to get an upper bound for the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k}{\mathbb E}_{s\in [p({\underline{h}})]} \left\Vert {\mathbb E}_{n\in [N^{ad}]}\, \, z_{N,{\underline{h}}}(n) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},n^{1/d})+e_{i,N}({\underline{h}},s)]} f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)}, \end{align*} $$

where $e_{i,N}({\underline {h}},s):=\frac {p_{i,r}({\underline {h}})}{p({\underline {h}})}s+\epsilon _{i,N}$ , $i=1,\ldots , \ell $ and $\epsilon _{1,N},\ldots , \epsilon _{\ell ,N}$ take finitely many values for $N\in {\mathbb N}$ . After replacing the average ${\mathbb E}_{s\in [p({\underline {h}})]}$ with $\max _{s\in [p({\underline {h}})]}$ , we are led to the asserted upper bound in equation (18).

Proof. We first reduce to the case where $e_{i,N}({\underline {h}})=0$ for $i=1, \ldots , \ell $ . To do this, we replace $[a_i({\underline {h}},n)+e_{i,N}({\underline {h}})]$ with $[a_i({\underline {h}},n)]+[e_{i,N}({\underline {h}})]$ ; this introduces some error sequences on the exponents that take finitely many values. To treat the error sequences, we use Lemma 3.6, redefine the weight $(c_{N,{\underline {h}}}(n))$ and introduce some sequences $\epsilon _{1,N}, \ldots , \epsilon _{\ell ,N}$ that take finitely many values for $N\in {\mathbb N}$ . Next, we compose with $T^{-[e_{1,N}({\underline {h}})]-\epsilon _{1,N}}$ , and we are left with upper-bounding the expression

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in [N^a]}\, c_{N,{\underline{h}}}(n) \cdot \prod_{i=1}^\ell T^{[a_i({\underline{h}},n)]} (T^{[e_{i,N}({\underline{h}})]-[e_{1,N}({\underline{h}})]+\epsilon_{i,N}-\epsilon_{1,N}}f_{N,{\underline{h}},i})\right\Vert_{L^2(\mu)}. \end{align*} $$

If we rename for $i=2,\ldots , \ell $ the functions $T^{[e_{i,N}({\underline {h}})]-[e_{1,N}({\underline {h}})]+\epsilon _{i,N}-\epsilon _{1,N}}f_{N,{\underline {h}},i}$ as $f_{N,{\underline {h}},i}$ , we are reduced to bounding equation (27) when $e_{i,N}({\underline {h}})=0$ for $i=1, \ldots , \ell $ .

We will prove the statement by induction on $\ell \in {\mathbb N}$ . For $\ell =1$ , the argument is similar to the one used in the inductive step, so we only summarise it briefly (for more details; see Steps 1–3 below). We first use Lemma 4.1, and we are led to upper-bounding the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in [N^a]}\, c_{N,{\underline{h}}}(n) \cdot T^{[p_1({\underline{h}})n+q_1({\underline{h}},n)]} f_{N,{\underline{h}},1}\right\Vert_{L^2(\mu)}, \end{align*} $$

where $p_1\neq 0$ and $q_1$ is a polynomial with real exponents and $\text {f-deg}(q_1)<1$ . We then apply equation (15) for the average over n, compose with $T^{-[p_1({\underline {h}})n+q_1({\underline {h}},n)]}$ , use that $q_1({\underline {h}},n+h_{k+1})-q_1({\underline {h}},n)$ is negligible for the range of parameters we are interested in and use Lemma 3.6 to treat the finite valued error sequences that arise. We get an upper bound by the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k, h_{k+1}\in [L_N]} \Big| \int T^{[p_1({\underline{h}})h_{k+1}]+\epsilon_N} f_{N,{\underline{h}},1} \cdot \overline{f}_{N,{\underline{h}},1} \, d\mu\Big|, \end{align*} $$

where $\epsilon _N$ takes finitely many values for $N\in {\mathbb N}$ . This proves equation (27) (with $\ell =r=1$ ).

Suppose that $\ell \geq 2$ and the statement holds for all nice collections of $\ell -1$ polynomials with real exponents and finitely many parameters.

We have that $a_i({\underline {h}},t):=\sum _{j=1}^r p_{i,j}({\underline {h}})t^{d_j}$ , $i=1,\ldots , \ell $ , where $0\leq d_1<\cdots <d_r=d\leq 1$ and $p_{i,j}\in {\mathbb R}[t_1,\ldots , t_k]$ . Furthermore, we can assume that the polynomial $p_{1,r}$ is nonzero, and hence the fractional degree of $a_1$ is d.

Step 1 (Linearising the highest-order term). If the fractional degree of $a_1$ is $1$ , then we proceed to Step 2. If not, then Lemma 4.1 (for $w_{N,{\underline {h}}}:=c_{N,{\underline {h}}}$ ) applies, and we get an estimate of the form equation (18). Hence, to get an estimate of the form equation (27), it suffices to get a similar estimate for the averages

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in [N^{ad}]}\, c_{N,{\underline{h}}}(n) \cdot \prod_{i=1}^{\ell} T^{[\tilde{a}_i({\underline{h}},n)]+e_{i,N}({\underline{h}})} f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)}, \end{align*} $$

where $(c_{N,{\underline {h}}}(n))$ is another $1$ -bounded sequence, $(e_{1,N}({\underline {h}})), \ldots , (e_{\ell ,N}({\underline {h}}))$ are sequences of real numbers and

(28) $$ \begin{align} \tilde{a}_i({\underline{h}},t):=p_{i,r}({\underline{h}})\,t+ q_i({\underline{h}},t), \quad \text{where} \quad q_i({\underline{h}},t):=\sum_{j=1}^{r-1} p_{i,j}({\underline{h}}) \, t^{\frac{d_j}{d}}, \quad i=1,\ldots, \ell. \end{align} $$

After composing with $T^{-e_{1,N}({\underline {h}})}$ and redefining the functions $f_{N,{\underline {h}},i}$ , $i=2,\ldots , \ell $ , we are reduced to the case where $e_{i,N}({\underline {h}})=0$ for $i=1,\ldots , \ell $ . So we only treat this case henceforth. We also remark that since the collection $a_1,\ldots , a_\ell $ is nice, and $\tilde {a}_i({\underline {h}},t)=a_i({\underline {h}},t^{1/d})$ , $i=1,\ldots , \ell $ , the collection $\tilde {a}_1,\ldots , \tilde {a}_\ell $ is also nice.

Step 2 (Reduction of $\ell $ via vdC). Applying equation (15) for the average over n, we get that it suffices to obtain an upper bound for the following averages:

$$ \begin{align*} {\mathbb E}_{({\underline{h}}, h_{k+1})\in [L_N]^{k+1}} {\mathbb E}_{n\in [N^{ad}]}\Big| \int \prod_{i=1}^\ell T^{[\tilde{a}_i({\underline{h}},n+h_{k+1})]} f_{N,{\underline{h}},i}\, \prod_{i=1}^\ell T^{[\tilde{a}_i({\underline{h}},n)]} \overline{f}_{N,{\underline{h}},i}\, \, d\mu\Big|. \end{align*} $$

We compose with $T^{-[\tilde {a}_1({\underline {h}},n)]}$ , and for $i=1,\ldots , \ell $ , we replace the differences $[\tilde {a}_i({\underline {h}},n+h_{k+1})]-[\tilde {a}_1({\underline {h}},n)]$ , $[\tilde {a}_i({\underline {h}},n)]-[\tilde {a}_1({\underline {h}},n)]$ with $[\tilde {a}_i({\underline {h}},n+h_{k+1})-\tilde {a}_1({\underline {h}},n)]$ , $[\tilde {a}_i({\underline {h}},n)-\tilde {a}_1({\underline {h}},n)]$ , respectively. To do so, we have to introduce some error sequences that take values on a finite subset of ${\mathbb N}$ . We use Lemma 3.6 to treat the errors that arise, and we are left with upper-bounding averages of the form

$$ \begin{align*} {\mathbb E}_{({\underline{h}}, h_{k+1})\in [L_N]^{k+1}} {\mathbb E}_{n\in [N^{ad}]}\Big| \int \prod_{i=1}^\ell T^{[\tilde{a}_i({\underline{h}},n+h_{k+1})-\tilde{a}_1({\underline{h}},n)]+\epsilon_{i,N}} &f_{N,{\underline{h}},i}\cdot \\ &\prod_{i=1}^\ell T^{[\tilde{a}_i({\underline{h}},n)-\tilde{a}_1({\underline{h}},n)]+\epsilon^{\prime}_{i,N} } \overline{f}_{N,{\underline{h}},i}\, \, d\mu\Big|, \end{align*} $$

where $\epsilon _{i,N},\epsilon ^{\prime }_{i,N}$ , $i=1,\ldots , \ell $ take finitely many values for $N\in {\mathbb N}$ . Note that the fractional degree of $q_1, \ldots , q_\ell $ is strictly smaller than $1$ . It follows from this and the mean value theorem that

(29) $$ \begin{align} \lim_{N\to\infty}\max_{({\underline{h}},h_k)\in [L_N]^{k+1}, i\in \{1,\ldots, \ell\}}|q_i({\underline{h}},t+h_{k+1})-q_i({\underline{h}},t)|=0. \end{align} $$

Using equations (28) and (29) and then Lemma 3.6, we get that it suffices to get an upper bound for the averages

$$ \begin{align*} {\mathbb E}_{({\underline{h}},h_{k+1})\in [L_N]^{k+1}} {\mathbb E}_{n\in [N^{ad}]}\, c_{N,{\underline{h}},h_{k+1}}(n) \cdot \int (T^{[p_{1,r}({\underline{h}})h_{k+1}]+\epsilon_{1,N} } &f_{N,{\underline{h}},1}\cdot \overline{f}_{N,{\underline{h}},1})\cdot \\ &\quad \prod_{i=2}^\ell T^{[b_i({\underline{h}},n)]+\epsilon_{i,N} }\tilde{f}_{N,{\underline{h}},h_{k+1},i}\, d\mu, \end{align*} $$

where $\epsilon _{1,N},\ldots , \epsilon _{\ell ,N}$ take finitely many values for $N\in {\mathbb N}$ ,

$$ \begin{align*} b_{i}({\underline{h}},t):=(p_{i,r}-p_{1,r})({\underline{h}})\, t+(q_{i}-q_1)({\underline{h}},t), \quad i=2,\ldots, \ell, \end{align*} $$

$\tilde {f}_{N,{\underline {h}},h_{k+1},i}\in L^\infty (\mu )$ , $i=2,\ldots , \ell $ are $1$ -bounded functions and $(c_{N,{\underline {h}},h_{k+1}}(n))$ is a $1$ -bounded sequence. Without loss of generality, we can assume that $b_\ell $ has maximal fractional degree within the collection $b_2,\ldots , b_\ell $ (note that some of the polynomials $p_{i,r}-p_{1,r}$ may vanish). We compose with $T^{-[b_\ell ({\underline {h}},n)]}$ and apply Lemma 3.6 to treat finite-valued error sequences that we get when we replace differences of integer parts with the integer part of the corresponding differences. After using the Cauchy-Schwarz inequality, we deduce that it suffices to get an upper bound for the following averages:

(30) $$ \begin{align} {\mathbb E}_{h_{k+1}\in [L_N]} \Big({\mathbb E}_{{\underline{h}}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in [N^{ad}]} \, c_{N,{\underline{h}},h_{k+1}}(n) \, \prod_{i=1}^{\ell-1} T^{[\tilde{b}_i({\underline{h}},n)]+\epsilon^{\prime}_{i,N}}\tilde{f}_{N,{\underline{h}},h_{k+1},i}\right\Vert_{L^2(\mu)}\Big), \end{align} $$

where $ \epsilon ^{\prime }_{1,N},\ldots , \epsilon ^{\prime }_{\ell -1,N}$ take finitely many values for $N\in {\mathbb N}$ ,

$$ \begin{align*} \tilde{b}_{i}({\underline{h}},t):=(p_{i,r}-p_{\ell,r})({\underline{h}})\, t+(q_{i}-q_\ell)({\underline{h}},t), \quad i=1,\ldots, \ell-1, \end{align*} $$

and

(31) $$ \begin{align} \tilde{f}_{N,{\underline{h}},h_{k+1},1}:= T^{[p_{1,r}({\underline{h}})h_{k+1}]+\epsilon_{1,N}}f_{N,{\underline{h}}, 1}\cdot \overline{f}_{N,{\underline{h}}, 1}, \end{align} $$

where $ \epsilon _{1,N}$ takes finitely many values for $N\in {\mathbb N}$ .

Note that our assumptions imply that $\tilde {b}_{1},\ldots , \tilde {b}_{\ell -1}$ , thought of as a collection of polynomials with real exponents and $(k+1)$ -parameters, is nice.

Step 3 (Applying the induction hypothesis). Using the induction hypothesis for the expression in equation (30) that is inside the parentheses, and the fact that $\tilde {b}_1,\ldots , \tilde {b}_{\ell -1}$ do not depend on the parameter $h_{k+1}$ , we get that there exist $l,r\in {\mathbb N}$ and nonconstant polynomials $P_1,\ldots , P_r\in {\mathbb R}[t_1,\ldots , t_{k+l}]$ with pairwise nonconstant differences, such that the averages in equation (30) are bounded by an $o_N(1)$ term plus a constant $C_{k,a_1,\ldots , a_\ell }$ (note that $\tilde {b}_{1},\ldots , \tilde {b}_{\ell -1}$ are determined by $a_1,\ldots , a_\ell $ ) times the expression

$$ \begin{align*} {\mathbb E}_{({\underline{h}}_1,h_{k+1})\in [L_N]^{k+1},{\underline{h}}_2\in [L_N]^l}\Big| \int \prod_{i=0}^{r} T^{[P_i({\underline{h}}_1,{\underline{h}}_2)]+\epsilon^{\prime}_{i,N}} F_{N,{\underline{h}}_1, h_{k+1},i}\, d\mu\Big|, \end{align*} $$

where $P_0:=0$ , $ F_{N,{\underline {h}},h_{k+1},i} \in \{ \tilde {f}_{N,{\underline {h}},h_{k+1},1}, \overline {\tilde {f}}_{N,{\underline {h}},h_{k+1},1} \}$ for $i=0,\ldots , r$ , ${\underline {h}}_1\in [L_N]^k$ , $h_{k+1}\in [L_N]$ , $N\in {\mathbb N}$ and $\epsilon ^{\prime }_{0,N},\ldots , \epsilon ^{\prime }_{r,N}$ take finitely many values for $N\in {\mathbb N}$ .

Using equation (31) and Lemma 3.6, we can bound this expression by a constant $C_{r}$ times the following average:

$$ \begin{align*} &{\mathbb E}_{({\underline{h}}_1,h_{k+1})\in [L_N]^{k+1},{\underline{h}}_2\in [L_N]^l} \Big|\int f_{N,{\underline{h}}_1, 1} \cdot T^{[p_{1,r}({\underline{h}}_1)h_{k+1}]+\epsilon^{\prime}_{0,N}}\overline{f}_{N,{\underline{h}}_1, 1}\cdot\\ &\qquad\qquad\qquad\qquad \prod_{i=1}^{r} \!\big(T^{[P_i({\underline{h}}_1,{\underline{h}}_2)+p_{1,r}({\underline{h}}_1)h_{k+1}]+\epsilon^{\prime}_{i,N}} G_{N,{\underline{h}}_1,h_{k+1},i}\cdot T^{[P_i({\underline{h}}_1,{\underline{h}}_2)]+\epsilon^{\prime}_{r+i,N}} G_{N,{\underline{h}}_1,h_{k+1},r+i}\!\big) d\mu\Big|, \end{align*} $$

where for $i=1,\ldots , 2r$ , we have $G_{N,{\underline {h}}_1,h_{k+1},i} \in \{ f_{N,{\underline {h}}_1,1}, \overline {f}_{N,{\underline {h}}_1,1} \}$ , ${\underline {h}}_1 \in [L_N]^k$ , $h_{k+1}\in [L_N]$ , $N\in {\mathbb N}$ , and $\epsilon ^{\prime }_{i,N}$ , $i=0,\ldots , 2r$ take finitely many values for $N\in {\mathbb N}$ . Since the polynomial $p_{1,r}$ is nonzero and the polynomials $P_1,\ldots , P_r$ with $k+l$ variables are nonconstant and have nonconstant pairwise differences, the same holds for the $2r+1$ polynomials with $k+l+1$ variables $p_{1,r}({\underline {h}}_1)h_{k+1}$ , $P_i({\underline {h}}_1,{\underline {h}}_2)+p_{1,r}({\underline {h}}_1)h_{k+1}$ , $P_i({\underline {h}}_1,{\underline {h}}_2)$ , $i=1,\ldots , r$ . This completes the proof.

Proof. The argument is similar to the one used to prove [Reference Leibman23, Theorem 1], where the case of polynomials with integer coefficients and $c_{N}({\underline {h}}):=1$ is covered, so we only sketch the points in the argument where one has to deviate slightly because of minor technical complications. The proof proceeds by induction on a certain vector, called the weight, that is associated to each polynomial family $P_1,\ldots , P_r$ in ${\mathbb R}[t_1,\ldots , t_k]$ .

The inductive step is carried out by using a variant of Lemma 3.5 in the form used in equation (16) that concerns averages over $[N]^k$ (see [Reference Leibman23, Lemma 4] for the precise statement). The argument applies verbatim in our case; the only change is that we need at various instances to replace the differences of the integer part of polynomials with the integer part of their differences; we do this with the help of Lemma 3.6, and the use of the constants $(c_N({\underline {h}}))$ facilitates this task.

The base case of the induction is the case where all the polynomials are linear with respect to all variables involved. This case is covered using another induction, this time on the number r of linear functions. The inductive step is proved using [Reference Leibman23, Lemma 4]. The only difference in our case, versus the argument used in [Reference Leibman23, Proposition 5], appears in the proof of the estimate

(32) $$ \begin{align} \limsup_{N\to\infty}{\mathbb E}_{{\underline{h}}\in [N]^k}\lvert\!|\!| g\cdot T^{[L({\underline{h}})]}f|\!|\!\rvert^{2^s}_{s}\leq C_L\, \lvert\!|\!| f|\!|\!\rvert^{2^{s+1}}_{s+1} \end{align} $$

for some $C_{L}>0$ , where $f,g\in L^\infty (\mu )$ and $L({\underline {h}})=\sum _{j=1}^k\alpha _j h_j$ for some $k\in {\mathbb N}$ and $\alpha _1,\ldots , \alpha _k\in {\mathbb R}$ . To obtain this bound, we first use Lemma 3.6 to show that it suffices to replace $[\sum _{j=1}^k\alpha _j h_j]$ with $\sum _{j=1}^k[\alpha _j h_j]$ , and we remark that the set

$$ \begin{align*} \{([\alpha_1 h_1],\ldots, [\alpha_k h_k])\colon (h_1,\ldots, h_k)\in {\mathbb N}^k\} \end{align*} $$

has bounded multiplicity and positive density (as a subset of ${\mathbb N}^k$ ). It follows that there exists $C_L>0$ such that

$$ \begin{align*} \limsup_{N\to\infty}{\mathbb E}_{{\underline{h}}\in [N]^k}\lvert\!|\!| g\cdot T^{\sum_{j=1}^k[\alpha_j h_j]}f|\!|\!\rvert^{2^s}_{s}\leq C_L\, \limsup_{N\to\infty}{\mathbb E}_{{\underline{h}}\in [N]^k}\lvert\!|\!| g\cdot T^{\sum_{j=1}^k h_j}f|\!|\!\rvert^{2^s}_{s}. \end{align*} $$

By [Reference Leibman23, Lemma 8], the last expression is bounded by a constant multiple of $\lvert \!|\!| f|\!|\!\rvert ^{2^{s+1}}_{s+1}$ . Combining the above, we get that equation (32) holds. Finally, the base case of the induction (of the linear case) is when $r=1$ and $P_1=L$ is linear. To cover this case, we again use [Reference Leibman23, Lemma 4] and reduce matters to the task of obtaining an upper bound for the expression

$$ \begin{align*} \limsup_{N\to\infty}{\mathbb E}_{{\underline{h}}\in [N]^k}\Big|\int \overline{f}\cdot T^{L({\underline{h}})}f\, d\mu\Big|. \end{align*} $$

By the $s=1$ case of equation (32) (recall that $\lvert \!|\!| f|\!|\!\rvert _1=|\int f\, d\mu |$ ), we get an upper bound by $C_L\lvert \!|\!| f|\!|\!\rvert _2^2$ for some $C_L>0$ . This completes the proof.

Proof. Combining Lemma 4.1 and Lemma 4.2 (for $f_{N,{\underline {h}},1}:=f_1$ , $N\in {\mathbb N}, {\underline {h}}\in {\mathbb N}^k$ ), we get that there exist $k,r\in {\mathbb N}$ and nonconstant polynomials $P_1,\ldots , P_r\in {\mathbb R}[t_1,\ldots , t_{k}]$ , with pairwise nonconstant differences, such that the averages in equation (8) are bounded by an $o_N(1)$ term plus a constant multiple of

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^{k}}\Big|\int \prod_{i=0}^{r} T^{[P_i({\underline{h}})]+ \epsilon_{i,N}} F_{i,{\underline{h}}}\, d\mu\Big|, \end{align*} $$

where $P_0:=0$ , $F_{0,{\underline {h}}},\ldots , F_{r,{\underline {h}}}\in \{f_1,\overline {f}_1\}$ , ${\underline {h}}\in {\mathbb N}^k$ and the sequences $\epsilon _{0,N},\ldots , \epsilon _{r,N}$ take values on a finite subset S of ${\mathbb Z}$ for $N\in {\mathbb N}$ . Since the limsup as $N\to \infty $ of the previous average is bounded by

$$ \begin{align*} \sum_{\epsilon_0,\ldots, \epsilon_r\in S,\, F_0,\ldots, F_r\in \{f_1,\overline{f}_1\}}\limsup_{N\to\infty}\Big({\mathbb E}_{{\underline{h}}\in [L_N]^{k}}\Big|\int \prod_{i=0}^{r} T^{[P_i({\underline{h}})]+ \epsilon_{i}} F_{ i}\, d\mu\Big|\Big), \end{align*} $$

it suffices to show that for all fixed $\epsilon _0,\ldots , \epsilon _r\in {\mathbb Z}$ and $F_{0},\ldots , F_{r}\in \{f_1,\overline {f}_1\}$ , we have

$$ \begin{align*} \lim_{N\to\infty}{\mathbb E}_{{\underline{h}}\in [L_N]^{k}}\Big|\int \prod_{i=0}^{r} T^{[P_i({\underline{h}})]+ \epsilon_{i}} F_{ i}\, d\mu\Big|=0. \end{align*} $$

The last average is equal to

$$ \begin{align*} {\mathbb E}_{{\underline{h}}\in [L_N]^{k}}\, c_N({\underline{h}})\cdot \int \prod_{i=0}^{r} T^{[P_i({\underline{h}})+\epsilon_i]} F_{ i}\, d\mu \end{align*} $$

for some $1$ -bounded sequence $(c_N({\underline {h}}))$ . The result now follows from Lemma 4.3.

Proof. Using the Taylor expansion of $a({\underline {h}},t)$ , we get that equation (33) holds with

$$ \begin{align*} e({\underline{h}},h_{k+1},t):= \frac{h_{k+1}^{d+1}}{(d+1)!}\, a^{(d+1)}({\underline{h}},\xi_{{\underline{h}},h_{k+1},t}) \end{align*} $$

for some $\xi _{{\underline {h}},h_{k+1},t}\in [t,t+h_{k+1}]$ . Since the fractional degree of a is $d+c$ for some $c\in (0,1)$ , we have

$$ \begin{align*} \max_{({\underline{h}},h_{k+1})\in [L_N]^{k+1}, t\in [\sqrt{N},N]}| e({\underline{h}},h_{k+1},t)|\prec \frac{L_N^A}{N^{\frac{1-c}{2}}} \end{align*} $$

for some $A>0$ that depends on d and the maximum degree of the coefficient polynomials of $a({\underline {h}},t)$ . Since $L_N\prec N^\varepsilon $ for every $\varepsilon>0$ , it follows that

$$ \begin{align*} \lim_{N\to\infty}\max_{({\underline{h}},h_{k+1})\in [L_N]^{k+1}, t\in [\sqrt{N},N]}| e({\underline{h}},h_{k+1},t)|=0, \end{align*} $$

completing the proof.

Proof. We first remark that if $\mathcal {A}$ consists of fractional polynomials with k-parameters and a is any fractional polynomial with k-parameters, then equation (34) implies that $\text {vdC}(\mathcal {A},a)$ consists of fractional polynomials with $(k+1)$ -parameters.

For $i=1,\ldots , \ell $ , let $\tilde {a}_i$ be the polynomial with real exponents and $(k+1)$ -parameters given by equation (34). We choose $a\in \mathcal {A}$ as follows:

1. 1. If $a_1,\ldots , a_\ell $ do not have the same fractional degree, we let $a_{i_0}$ be a function in the family $\{a_2,\ldots , a_\ell \}$ that has minimal (positive) fractional degree and set $a=a_{i_0}$ .

2. 2. If $a_1,\ldots , a_\ell $ have the same fractional degree, we let $i_0\in \{1,\ldots , \ell \}$ be so that $\tilde {a}_1-a_{i_0}$ has maximal degree within the family $\tilde {a}_1-a_1, \ldots , \tilde {a}_1-a_\ell $ and set $a=a_{i_0}$ .

By construction, all functions in $\text {vdC}(\mathcal {A},a)$ are nonconstant (we have removed constant functions). We first show that independently of the choice of a, the difference of $\tilde {a}_1-a$ with a function in $\text {vdC}(\mathcal {A},a)$ is always nonconstant (in the variable t); in the process, we also show that $\text {f-deg}(\tilde {a}_1-a)>0$ . Suppose that such a difference has the form $\tilde {a}_1-a_i$ for some $i\in \{1, \ldots , \ell \}$ . It follows from Lemma 5.1 that $\tilde {a}_1$ contains the term $h_{k+1}a_1'(t)$ , which depends nontrivially on the parameter $h_{k+1}$ (note also that $a_1,\ldots , a_\ell $ do not depend on this parameter). It follows from this and our assumption $\text {f-deg}(a_1)>1$ that

$$ \begin{align*}\text{f-deg}(\tilde{a_1}-a_i)\geq \text{f-deg}(a_1')=\text{f-deg}(a_1)-1>0, \quad i=1,\ldots, \ell. \end{align*} $$

It remains to cover the case where the difference of $\tilde {a}_1-a$ with a function in $\text {vdC}(\mathcal {A},a)$ has the form $\tilde {a}_1-\tilde {a}_i$ for some $i\in \{2, \ldots , \ell \}$ . Then using Lemma 5.1 and our assumption that $\mathcal {A}$ is nice, we get

$$ \begin{align*}\text{f-deg}(\tilde{a}_1-\tilde{a}_i)\geq \text{f-deg}(a_1-a_i)>0, \quad i=2,\ldots, \ell. \end{align*} $$

Next we show that $\tilde {a}_1-a$ has maximal fractional degree within the family $\text {vdC}(\mathcal {A},a)$ . Suppose first that we are in Case $(i)$ . Since $\text {f-deg}(a_{i_0})<\text {f-deg}(a_1)$ , we have that $\tilde {a}_1-a_{i_0}$ has the same fractional degree as $a_1$ , which by assumption has maximal fractional degree within the family $\{a_1,\ldots , a_\ell \}$ . We deduce that $\tilde {a}_1-a_{i_0}$ has maximal fractional degree within the family $\text {vdC}(\mathcal {A},a)$ . Suppose now that we are in Case $(ii)$ and let $i\in \{1,\ldots , \ell \}$ . Since $a_i-a_{i_0}=(a_i-\tilde {a}_1)+(\tilde {a}_1-a_{i_0})$ and by the choice of $i_0$ we have $\text {f-deg}(\tilde {a}_1-a_{i_0})\geq \text {f-deg}(\tilde {a}_1-a_i)$ , we deduce that

(35) $$ \begin{align} \text{f-deg}(\tilde{a}_1-a_{i_0})\geq \text{f-deg}(a_i-a_{i_0}), \quad i=1, \ldots, \ell. \end{align} $$

Moreover, note that $\tilde {a}_i-a_{i_0}=(\tilde {a}_i-a_i)+(a_i-a_{i_0})$ and

(36) $$ \begin{align} \text{f-deg}(\tilde{a}_1-a_{i_0})\geq \text{f-deg}(\tilde{a}_1-a_1)=\text{f-deg}(a_1)-1\geq\text{f-deg}(a_i)-1=\text{f-deg}(\tilde{a}_i-a_i), \end{align} $$

where the two identities follow from Lemma 5.1, and the first estimate follows from the choice of $i_0$ and the second since the family $\mathcal {A}$ is nice. We deduce from equations (35) and (36) that

(37) $$ \begin{align} \text{f-deg}(\tilde{a}_1-a_{i_0})\geq \text{f-deg}(\tilde{a}_i-a_{i_0}), \quad i=1,\ldots, \ell. \end{align} $$

Combining equations (35) and (37), we get that $\tilde {a}_1-a_{i_0}$ has maximal fractional degree within the family $\text {vdC}(\mathcal {A},a)$ .

Using Lemma 5.1 and the definition of the degree, it is easy to verify that if for some $i\in \{1,\ldots , \ell \}$ , we have $a_i\not \cong a_{i_0}$ , then $\deg (a_i-a_{i_0})=\deg (\tilde {a}_i-a_{i_0})=\deg (a_i)$ and $a_i-a_{i_0}\cong \tilde {a}_i-a_{i_0}$ , while if $a_i\cong a_{i_0}$ , then $\deg (a_i-a_{i_0})<\deg (a_i)$ and $\deg (\tilde {a}_i-a_{i_0})<\deg (a_i)$ . Using these facts, we easily get the following:

If we are in Case $(i)$ , we have that the type of $\mathcal {A}$ has the form $(d, k_d, \ldots , k_l,0,\ldots , 0)$ , where $l=\deg (a_{i_0})$ , $k_l\geq 1$ , and $d\geq 1$ . Then the type of $\text {vdC}(\mathcal {A},a)$ is $(d, k_d, \ldots , k_l-1)$ if $l=0$ , and $(d, k_d, \ldots , k_l-1,k_{l-1},\ldots , k_0)$ for some $k_0,\ldots , k_{l-1}\in {\mathbb Z}_+$ if $l\geq 1$ .

If we are in Case $(ii)$ , we have that the type of $\mathcal {A}$ has the form $(d, k_d,0, \ldots , 0)$ , where $d\geq 1$ and $k_d\geq 1$ . Then for every $a\in \mathcal {A}$ , the type of $\text {vdC}(\mathcal {A},a)$ is $(d, k_d-1,k_{d-1}\ldots , k_0)$ for some $k_0,\ldots , k_{d-1}\in {\mathbb Z}_+$ .

In both cases, the type of the family $\text {vdC}(\mathcal {A},a)$ is smaller than the type of the family $\mathcal {A}$ , completing the proof of Claim 2.

Proof of Theorem 3.1

Our goal is to show that there exists $s\in {\mathbb N}$ such that if $f_{N,{\underline {h}},1}=f_1$ , ${\underline {h}}\in [L_N]^k,N\in {\mathbb N}$ , $\lvert \!|\!| f_1|\!|\!\rvert _s=0$ and all other functions below are assumed to be $1$ -bounded, then

$$ \begin{align*}\lim_{N\to\infty} {\mathbb E}_{\underline{h}\in [L_N]^k} \left\Vert {\mathbb E}_{n\in [N]}\, w_{N,\underline{h}}(n)\cdot \prod_{i=1}^\ell T^{[a_i(\underline{h},n)]}f_{N,{\underline{h}},i}\right\Vert_{L^2(\mu)}=0, \end{align*} $$

where $w_{N,{\underline {h}}}(n):=(\Delta _{\underline {h}}\Lambda ')(n)\cdot c_{N,{\underline {h}}}(n)$ , $ {\underline {h}}\in [L_N]^k, n\in [N], N\in {\mathbb N}$ and the sequence $(c_{N,{\underline {h}}}(n))$ is $1$ -bounded.

We prove this using induction on the type of the nice family of fractional polynomials $\mathcal {A}:=\{a_1,\ldots , a_\ell \}$ with finitely many parameters. If $\text {f-deg}(a_1)<1$ (then also $\text {f-deg}(a_j)<1$ for $j=2,\ldots , \ell $ ), then the result follows from Proposition 4.4.

Suppose that the family $\mathcal {A}:=\{a_1,\ldots , a_\ell \}$ has type $(d,k_d,\ldots , k_0)$ , where $d\geq 1$ , $k_d\geq 1$ , $k_{d-1},\ldots , k_0\in {\mathbb Z}_+$ , and the statement holds for all families of fractional polynomials with finitely many parameters and type strictly smaller than $(d,k_d,\ldots , k_0)$ . Since $\deg (a_1)\geq 1$ and $a_1$ is a fractional polynomial, we have that $\text {f-deg}(a_1)>1$ .

By Lemma 5.2, there exists $a\in \mathcal {A}$ such that the family $\text {vdC}(\mathcal {A},a)$ , ordered so that the first function is $\tilde {a}_1-a$ (where $\tilde {a}_1$ is as in equation (34)), consists of fractional polynomials with finitely many parameters and satisfies the following:

(38) $$ \begin{align} \text{vdC}(\mathcal{A},a) \quad \text{is nice and has type strictly smaller than} \quad (d,k_d,\ldots, k_0). \end{align} $$

We use equation (15) for the average ${\mathbb E}_{n\in [N]}$ , compose with $T^{-[a({\underline {h}},n)]}$ and then use the Cauchy-Schwarz inequality. We get that it suffices to show the following (recall that $(\Delta _h u)(n)=u(n+h)\cdot \overline {u(n)}$ ):

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{(\underline{h},h_{k+1})\in [L_N]^{k+1}} \Big|\Big|{\mathbb E}_{n\in [N]}\, (\Delta_{h_{k+1}}w_{N,\underline{h}})(n)\cdot \prod_{i=1}^\ell &T^{[a_i(\underline{h},n+h_{k+1})]-[a({\underline{h}},n)]}f_{N,{\underline{h}},i}\cdot \\ &\qquad \ \prod_{i=1}^\ell T^{[a_i(\underline{h},n)]-[a({\underline{h}},n)]}\overline{f}_{N,{\underline{h}},i}\Big|\Big|_{L^2(\mu)}=0. \end{align*} $$

We replace the differences of integer parts on the iterates with the integer part of their differences and also replace $a_i({\underline {h}},n+h_{k+1})$ with $\tilde {a}_i({\underline {h}}, h_{k+1}, n)$ , where $\tilde {a}_j$ is associated to $a_j$ by equation (34) of Lemma 5.1. To make these substitutions, we introduce some error sequences that take finitely many values; as usual, these sequences can be handled after we apply Lemma 3.6 (which applies without a problem since the values of n that are smaller than $\sqrt {N}$ contribute negligibly in the average). After completing these maneuvers, we see that it suffices to show the following:

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{({\underline{h}},h_{k+1})\in [L_N]^{k+1}} \left\Vert {\mathbb E}_{n\in [N]}\, w_{N,{\underline{h}},h_{k+1}}(n) \cdot \prod_{i=1}^{2\ell} T^{[b_i({\underline{h}},h_{k+1},n)]+\epsilon_{i,N}}g_{N,{\underline{h}},h_{k+1},i}\right\Vert_{L^2(\mu)}=0, \end{align*} $$

where $\epsilon _{1,N},\ldots , \epsilon _{2\ell ,N}$ take finitely many values for $N\in {\mathbb N}$ ,

$$ \begin{align*}w_N({\underline{h}}, h_{k+1},n):= (\Delta_{({\underline{h}},h_{k+1})}\Lambda')(n)\cdot c_{N,{\underline{h}}, h_{k+1}}(n) \end{align*} $$

for some $1$ -bounded sequence $(c_{N,{\underline {h}}, h_{k+1}}(n))$ and

$$ \begin{align*} b_i({\underline{h}}, h_{k+1},t)&:=\tilde{a}_i({\underline{h}}, t+h_{k+1})-a({\underline{h}},t),\quad i=1,\ldots, \ell,\\ b_{\ell+i}({\underline{h}}, h_{k+1},t)&:=a_i({\underline{h}}, t)-a({\underline{h}},t), \quad i=1,\ldots, \ell \end{align*} $$

and $g_{N,{\underline {h}},h_{k+1},i}$ are $1$ -bounded functions in $L^\infty (\mu )$ such that $g_{N,{\underline {h}},h_{k+1},1}:=f_1$ for all $({\underline {h}},h_{k+1})\in [L_N]^{k+1}$ , $N\in {\mathbb N}$ . We compose with $T^{-\epsilon _{1,N}}$ inside the $L^2(\mu )$ -norm and set $h_{N,{\underline {h}},h_{k+1},i}:=T^{\epsilon _{i,N}-\epsilon _{1,N}}g_{N,{\underline {h}},h_{k+1},i}$ , $i=1,\ldots , 2\ell $ (then $h_{N,{\underline {h}},h_{k+1},1}=f_1$ ). We get that it suffices to show that

(39) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{({\underline{h}},h_{k+1})\in [L_N]^{k+1}} \left\Vert {\mathbb E}_{n\in [N]}\, w_{N,{\underline{h}},h_{k+1}}(n) \cdot \prod_{i=1}^{2\ell} T^{[b_i({\underline{h}},h_{k+1},n)]}h_{N,{\underline{h}},h_{k+1},i}\right\Vert_{L^2(\mu)}=0. \end{align} $$

Finally, we can remove all functions associated with iterates that do not depend on the variable n (note that by Lemma 5.2, the function $b_1$ is not one of them), and thus we arrive at an average with iterates given by the family $\text {vdC}(\mathcal {A},a)$ , ordered so that the first function is $\tilde {a}_1-a$ . By the choice of a, we have that equation (38) holds. Hence, the induction hypothesis applies for this family and gives that there exists $s\in {\mathbb N}$ such that if $\lvert \!|\!| f_1|\!|\!\rvert _s=0$ , then equation (39) holds. This completes the induction step and the proof.