Proof. The claims (i) all follow easily the triangle inequality and the observation that the intersection of two arithmetic progressions is again an arithmetic progression; for instance, equation (2.3) follows from the observation that any subprogression $P'$ of P is partitioned into subprogressions $P' \cap P_1, P' \cap P_2$ of $P_1, P_2$ , respectively. To prove (ii), we observe from (i) that for any $0 < t < H/2$ we have

$$ \begin{align*} {\left|\sum_{x_0 < n \leq x_0+H} f(n)\right|}^* &\leq {\left|\sum_{x_0 < n \leq x_0+H/2} f(n)\right|}^* + {\left|\sum_{x_0+H/2 < n \leq x_0+H} f(n)\right|}^*\\ &\leq {\left|\sum_{x_0-t < n \leq x_0-t+H} f(n)\right|}^* + {\left|\sum_{x_0+t < n \leq x_0+t+H} f(n)\right|}^* \end{align*} $$

and the claim then follows by averaging in t.

To prove the first claim (2.4) of (iii), it will suffice by the monotonicity properties of total variation and maximal sums to show that

(2.6) $$ \begin{align} {\left| \sum_{n \in P'} f(n) g(n) \right| \leq \|g\|_{{\operatorname{TV}}(P')} \left| \sum_{n \in P'} f(n) \right|}^* \end{align} $$

for all subprogressions $P'$ of P. Clearly, we may assume $P'$ is nonempty. If we order the elements of $P'$ as $n_1 < n_2 < \dots < n_k$ , then from summation by parts we have

$$ \begin{align*}\sum_{n \in P'} f(n) g(n) = \sum_{j=1}^{k-1} (g(n_j) - g(n_{j+1})) \sum_{i=1}^j f(n_i) + g(n_k) \sum_{i=1}^k f(n_i).\end{align*} $$

Since each segment $\{n_1,\dots ,n_j\}$ of $P'$ is again a subprogression of $P'$ , we have from the triangle inequality that

$$ \begin{align*}\left|\sum_{n \in P'} f(n) g(n)\right| \leq \sum_{j=1}^{k-1} |g(n_j) - g(n_{j+1})| {\left| \sum_{n \in P'} f(n) \right|}^* + |g(n_k)| {\left| \sum_{n \in P'} f(n) \right|}^*\end{align*} $$

and the claim (2.6) now follows from Definition 2.1. Thus, equation (2.4) holds. To prove the second claim (2.5), partition P into subprogressions $P \cap (a+q\mathbb {Z})$ , apply equation (2.4) to each subprogression and sum using (i).

Proof. By applying a translation, we may assume that I takes the form $(0,N]$ for some $N \geq 1$ . We may also assume $\varepsilon \leq \delta /2$ , since we are clearly done otherwise. We may now invoke [Reference Green and Tao19, Lemma 4.5] to conclude that there exists $1 \leq q \ll _d \delta ^{-O_d(1)} \varepsilon $ such that

(2.7) $$ \begin{align} \sup_{1 \leq j \leq d} \sup_{n \in I} |I|^j \| q \partial^j_1 P(n) \|_{\mathbb{R}/\mathbb{Z}} \ll_d \delta^{-O_d(1)} \varepsilon. \end{align} $$

This is almost what we want, except that we have to also control the $j=0$ contribution. But from hypothesis, we have at least one $n_0 \in I$ such that $\|P(n_0)\|_{\mathbb {R}/\mathbb {Z}} \leq \varepsilon $ , and from equation (2.7) we have $\| q \partial _1 P(n) \|_{\mathbb {R}/\mathbb {Z}} \ll _d \delta ^{-O_d(1)} |I|^{-1} \varepsilon $ for all $n \in I$ . From the triangle inequality, we then conclude that

$$ \begin{align*}\| qP(n) \|_{\mathbb{R}/\mathbb{Z}} \ll_d \delta^{-O_d(1)} \varepsilon\end{align*} $$

for all $n \in I$ , and the claim follows.

Proof. We allow all implied constants to depend on d. We may assume that $|I| \geq C \delta ^{-C} A$ for a large constant C depending on d, as the claim is immediate otherwise.

We now claim that for each $0 \leq j \leq d$ that there exists a decomposition

(2.9) $$ \begin{align} P = P_j + Q_j, \end{align} $$

where $P_j \colon \mathbb {Z} \to \mathbb {R}/\mathbb {Z}$ is a polynomial of degree at most d with

(2.10) $$ \begin{align} \| q_j P_j \|_{C^\infty(I)} \ll \delta^{-O(1)} \end{align} $$

for some $1 \leq q_j \ll \delta ^{-O(1)}$ , and $Q_j \colon \mathbb {Z} \to \mathbb {R}/\mathbb {Z}$ is a polynomial of degree at most j. For $j=d$ , one can simply set $P_d=0$ and $Q_d=P$ . Now, suppose by downward induction that $0 \leq j < d$ and the claim has already been proven for $j+1$ . From equation (2.10) (for $P_{j+1}$ ), we have

$$ \begin{align*}\| q_{j+1} P_{j+1} \|_{C^\infty(I)} \ll \delta^{-O(1)}.\end{align*} $$

Routine Taylor expansion then gives

$$ \begin{align*}\| q_{j+1} P_{j+1}(a \cdot) \|_{C^\infty(\frac{1}{a} I)} \ll \delta^{-O(1)}\end{align*} $$

for all $a \in [A,2A]$ , thus by equation (2.8) and the triangle inequality we have

$$ \begin{align*}\| q_{j+1} Q_{j+1}(a \cdot) \|_{C^\infty(\frac{1}{a} I)} \ll \delta^{-O(1)}\end{align*} $$

for $\geq \delta A$ choices of $a \in [A,2A]$ .

Now, write $Q_{j+1}(n) = \alpha _{j+1} \binom {n}{j+1} + Q_j(n)$ , where $Q_j$ is of degree at most j. Taking $j+1$ -fold derivatives, we see that

$$ \begin{align*}\| a^{j+1} q_{j+1} \alpha_{j+1} \|_{\mathbb{R}/\mathbb{Z}} \ll \delta^{-O(1)} (A/|I|)^{j+1}\end{align*} $$

for $\geq \delta A$ choices of $a \in [A,2A]$ . Applying Lemma 2.3 to the polynomial $a \to a^{j+1}q_{j+1}\alpha _{j+1}$ (and recalling that $|I|/A \geq C \delta ^{-C}$ for a suitably large C by assumption), we conclude that there is $1 \leq q \ll \delta ^{O(1)}$ such that

$$ \begin{align*}\| q (\cdot)^{j+1} q_{j+1} \alpha_{j+1} \|_{C^\infty([A,2A])} \ll \delta^{-O(1)} (A/|I|)^{j+1}\end{align*} $$

and hence on taking $j+1$ -fold derivatives

$$ \begin{align*}\| (j+1)! q q_{j+1} \alpha_{j+1} \|_{\mathbb{R}/\mathbb{Z}} \ll \delta^{-O(1)} |I|^{-j-1}.\end{align*} $$

If one then sets and , we obtain the decomposition (2.9), and equation (2.10) follows from the triangle inequality. This closes the induction. Applying the claim with $j=0$ , we obtain the corollary.

Proof. By applying a translation, we may assume $I = (0,N]$ for some $N \geq 1$ . The claim now follows from [Reference Shao and Teräväinen59, Theorem 3.5].

Proof. We allow all implied constants to depend on $d,D$ . Let $\tilde \pi \colon \log G \to \log G / \log N \equiv \log (G/N)$ be the quotient map of $\log G$ by the Lie algebra ideal $\log N$ , then $\pi \circ \exp = \exp \circ \tilde \pi $ . For each $0 \leq i \leq d$ , the vectors $\tilde \pi (X_j)$ for $D - \dim G_i < j \leq D$ span the linear subspace $\tilde \pi (\log G_i)$ of $\log (G/N)$ , and the linear relations between those vectors are are generated by $O(1)$ equations with coefficients rational of height $O(\delta ^{-O(1)})$ . From this and linear algebra, we may find a basis $\tilde X_1,\dots ,\tilde X_{\dim (G/N)}$ of $\log (G/N)$ such that for each $0 \leq i \leq d$ , $\tilde \pi (\log G_i)$ is the span of $\tilde X_j$ for $\dim (G/N) - \dim \tilde \pi (\log G_i) < j \leq \dim (G/N)$ , and each $\tilde X_j$ is a linear combination of the $\tilde \pi (X_1),\dots ,\tilde \pi (X_D)$ with coefficients rational of height $O(\delta ^{-O(1)})$ . Meanwhile, $\pi (\Gamma )$ is generated by $\pi (X_1),\dots ,\pi (X_D)$ . From this and the Baker–Campbell–Hausdorff formula, we see that the basis $\tilde X_1,\dots ,\tilde X_{\dim (G/N)}$ is a $O(\delta ^{-O(1)})$ -rational weak basis for $\pi (G)/\pi (\Gamma )$ in the sense of [Reference Green and Tao19, Definition A.7]. Applying [Reference Green and Tao19, Proposition A.9] to this weak basis, we obtain a Mal’cev basis that gives $\pi (G)/\pi (\Gamma )$ the structure of a filtered nilmanifold with the stated degree, dimension and complexity. It remains to establish the bound (2.11). By right translation invariance, we can take g to be the identity. For the upper bound, it suffices (since $\pi $ is N-invariant) to show that

$$ \begin{align*}d_{\pi(G)}(1, \pi(h) ) \ll \delta^{-O(1)} d_G(1, h),\end{align*} $$

but this follows from the fact that $\tilde \pi \colon \log G \to \tilde \pi (\log G)$ has operator norm $O(\delta ^{-O(1)})$ when using the $X_1,\dots ,X_D$ basis for $\log G$ and the $\tilde X_1,\dots ,\tilde X_{\dim (G/N)}$ basis for $\tilde \pi (\log G)$ to define norms.

Now, we need to establish the lower bound. By [Reference Green and Tao19, Lemma A.4], it suffices to show that

$$ \begin{align*}\| Y \| \gg \delta^{-O(1)} \inf_{Y' \in \tilde \pi^{-1}(Y)} \|Y'\|\end{align*} $$

for any $Y \in \tilde \pi (\log G)$ , where again we use the norm given by the $X_1,\dots ,X_D$ basis for $\log G$ and the $\tilde X_1,\dots ,\tilde X_{\dim (G/N)}$ . But this is easily verified for each $Y = \tilde X_i$ , and the claim then follows by linearity.

Proof. We allow all implied constants to depend on $d,D$ . Since $Z(G) / (Z(G) \cap \Gamma )$ is an abelian filtered nilmanifold of complexity $O(\delta ^{-O(1)})$ , it can be identified with a torus $\mathbb {R}^m/\mathbb {Z}^m$ , where $m=O(1)$ and the metric on $Z(G)$ is comparable to the metric on $\mathbb {R}^m$ up to factors of $O(\delta ^{-O(1)})$ ; the identification of $\log Z(G)$ with $\mathbb {R}^m$ induces a logarithm map $\log \colon Z(G) \to \mathbb {R}^m$ and an exponential map $\exp \colon \mathbb {R}^m \to Z(G)$ . Central frequencies $\xi $ can then be identified with elements $k_\xi $ of $\mathbb {Z}^m$ , with $\xi (z) = k_\xi \cdot \log (z)$ for any $z \in Z(G)$ .

Let $\varphi : \mathbb {R}^m \to \mathbb {R}$ be a fixed bump function (depending only on m) that equals $1$ at the origin, and let $R>1$ be a parameter to be chosen later. For any central frequency $\xi $ , we set

where $dz$ is Haar probability measure on the torus $\mathbb {R}^m/\mathbb {Z}^m$ , which acts centrally on $G/\Gamma $ in the obvious fashion. It is easy to see that $F_\xi $ has Lipschitz norm $O(\delta ^{-O(1)})$ , oscillates with central frequency $\xi $ , and vanishes unless $\xi $ has Lipschitz norm $O( \delta ^{-O(1)} R^{O(1)} )$ ; also, if F has mean zero, then so do all of the $F_\xi $ . From the Fourier inversion formula, we have

$$ \begin{align*}\varphi(k_\xi/R) = \int_{\mathbb{R}^m} \hat \varphi(y) e( k_\xi \cdot y/R )\ dy = \int_{\mathbb{R}^m} \hat \varphi(y) e( \xi(\exp(y/R)) )\ dy ,\end{align*} $$

where

, as well as the Fourier inversion formula on the torus,

$$ \begin{align*}\sum_\xi F_\xi(x) = \int_{\mathbb{R}^m} \hat \varphi(y) F( \exp(y/R) x )\ dy.\end{align*} $$

On the other hand, from the Lipschitz nature of F we have

$$ \begin{align*}F( \exp(y/R) x ) = F(x) + O( \delta^{-O(1)} |y| / R ).\end{align*} $$

Since $\hat \varphi $ is rapidly decreasing and has total integral $1$ , we obtain

$$ \begin{align*}F = \sum_\xi F_\xi + O(\delta^{-O(1)}/R),\end{align*} $$

and the claim follows by choosing $R = O(\delta ^{-O(1)})$ suitably.

Proof. This is a slight variant of [Reference Green and Tao19, Lemma 7.9], the main difference being that our hypothesis (2.12) involves $\eta \circ g$ rather than $\eta \circ g_2$ (where $g_2$ is the nonlinear part of g). The argument in the proof of [Reference Green and Tao19, Lemma 7.9] can be modified in an obvious manner as follows. By translation, we may assume that $I = [1, |I|]$ . Let $\psi : G\rightarrow \mathbb {R}^D$ be the Mal’cev coordinate map. Suppose that

$$ \begin{align*}\psi(g(n)) = t_0 + \binom{n}{1}t_1 + \binom{n}{2}t_2 + \cdots + \binom{n}{d}t_d\end{align*} $$

for some $t_0, t_1,\cdots ,t_d \in \mathbb {R}^D$ with $\psi ^{-1}(t_i) \in G_i$ . Our assumption on $\|\eta \circ g\|_{C^{\infty }(I)}$ implies that for some $k \in \mathbb {Z}^D$ with $|k| \leq \delta ^{-1}$ , we have

$$ \begin{align*}\|k \cdot t_i\|_{\mathbb{R}/\mathbb{Z}} \ll \delta^{-O_{d,D}(1)}/|I|\end{align*} $$

for each $1 \leq i \leq d$ . Choose $u_i \in \mathbb {R}^D$ with $\psi ^{-1}(u_i) \in G_i$ such that

$$ \begin{align*}k \cdot u_i \in \mathbb{Z}, \ \ |t_i - u_i| \ll \delta^{-O_{d,D}(1)}/|I|.\end{align*} $$

Then choose $v_i \in \mathbb {R}^D$ with $\psi ^{-1}(v_i) \in G_i$ , all of whose coordinates are rationals over some denominator $\ll \delta ^{-O_{d,D}(1)}$ , such that

$$ \begin{align*}k \cdot u_i = k \cdot v_i\end{align*} $$

for each $1 \leq i \leq d$ . Define $\varepsilon ,\gamma $ by

$$ \begin{align*}\psi(\varepsilon(n)) = t_0 + \sum_{i=1}^d \binom{n}{i}(t_i - u_i), \ \ \psi(\gamma(n)) = \sum_{i=1}^d \binom{n}{i}v_i,\end{align*} $$

and then define $g'$ by

$$ \begin{align*}g'(n) = \varepsilon(n)^{-1}g(n) \gamma(n)^{-1}.\end{align*} $$

One can verify that they satisfy the desired properties.

Proof. See [Reference Green and Tao19, Theorem 1.19] (after rounding I to integer endpoints and translating to be of the form $[1,N]$ ). The additional requirement in (ii) that one has equidistribution in the larger nilmanifolds $G'/\Gamma "$ is not stated in [Reference Green and Tao19, Theorem 1.19] but follows easily from the proof, the point being that if a sequence $g' \in {\operatorname {Poly}}(\mathbb {Z} \to G')$ fails to be totally $1/M^A$ -equidistributed in $G'/\Gamma "$ , then one has $\|\eta \circ g' \|_{C^\infty (I)} \ll _{d,D} M^{O_{d,D}(A)}$ for some nontrivial horizontal character $\eta $ on $G'/\Gamma "$ of Lipschitz norm $O_{d,D}(M^{O_{d,D}(A)})$ , which on multiplying $\eta $ by the index of $\Gamma "$ in $\Gamma '$ also gives $\|\eta ' \circ g' \|_{C^\infty (I)} \ll _{d,D} M^{O_{d,D}(A)}$ for some nontrivial horizontal character $\eta '$ on $G'/\Gamma '$ of Lipschitz norm $O_{d,D}(M^{O_{d,D}(A)})$ . As a consequence, one can replace all occurrences of $G'/\Gamma '$ in the proof of [Reference Green and Tao19, Theorem 1.19] with $G'/\Gamma "$ with only negligible changes to the arguments.

Proof. This follows from [Reference Green and Tao19, Theorem 10.2], after implementing the corrections in [Reference Green and Tao20], and the modifications indicated in the proof of Theorem 2.12.

Proof. We allow all implied constants to depend on $d,D$ . The product of the filtered nilmanifold $G/\Gamma $ with itself is again a filtered nilmanifold $(G \times G)/(\Gamma \times \Gamma )$ , with the obvious filtration

and Mal’cev basis $(X_i,0), (0,X_i)$ , $i=1,\dots ,D$ . This product filtered nilmanifold has degree at most d, dimension $2D$ and complexity at most $O(\delta ^{-O(1)})$ . The pair $(g_1,g_2)$ can be then viewed as an element of ${\operatorname {Poly}}(\mathbb {Z} \to G \times G)$ . If we let $F \otimes \overline {F} \colon (G \times G)/(\Gamma \times \Gamma ) \to \mathbb {C}$ be the function

then F is Lipschitz with norm $O( \delta ^{-O(1)})$ and one has

(2.13) $$ \begin{align} {\left|\sum_{n \in I} F \otimes \overline{F}((g_1,g_2)(n) (\Gamma \times \Gamma))\right|}^* \geq \delta |I|. \end{align} $$

Let $A>1$ be sufficiently large depending on $d,D$ . Applying Theorem 2.12 to $(g_1,g_2)$ (with $\delta $ replaced by $\delta ^A$ ), we can find $\delta ^{-A} \leq M \ll _A \delta ^{-O_A(1)}$ and a factorization

(2.14) $$ \begin{align} (g_1,g_2) = (\varepsilon_1,\varepsilon_2) (g^{\prime}_1, g^{\prime}_2) (\gamma_1,\gamma_2) \end{align} $$

where $\varepsilon _1, g^{\prime }_1, \gamma _1 \in {\operatorname {Poly}}(\mathbb {Z} \to G_1)$ , $\varepsilon _2, g^{\prime }_2, \gamma _2 \in {\operatorname {Poly}}(\mathbb {Z} \to G_2)$ such that

1. (i) $(\varepsilon _1,\varepsilon _2)$ is $(M,I)$ -smooth;

2. (ii) There is an M-rational subnilmanifold $G'/\Gamma '$ of $(G \times G)/(\Gamma \times \Gamma )$ such that $(g^{\prime }_1,g^{\prime }_2)$ takes values in $G'$ and is totally $1/M^A$ -equidistributed in $G'/\Gamma "$ for any subgroup $\Gamma "$ of $\Gamma '$ of index at most $M^A$ ; and

3. (iii) $(\gamma _1,\gamma _2)$ is M-rational.

We caution that $G'$ is a subgroup of $G \times G$ rather than G. From equation (2.13), we thus have

$$ \begin{align*}{\left|\sum_{n \in I} F \otimes \overline{F}( (\varepsilon_1,\varepsilon_2)(n) (g^{\prime}_1,g^{\prime}_2)(n) (\gamma_1,\gamma_2)(n) (\Gamma \times \Gamma))\right|}^* \geq \delta |I|.\end{align*} $$

Since $(\gamma _1,\gamma _2)$ is M-rational, it is $O(M^{O(1)})$ -periodic and then by the pigeonhole principle (and Lemma 2.2(i)) we can thus find M-rational $(\gamma _1^0, \gamma _2^0) \in G \times G$ such that

$$ \begin{align*}{\left |\sum_{n \in I} F \otimes \overline{F}( (\varepsilon_1,\varepsilon_2)(n) (g^{\prime}_1,g^{\prime}_2)(n) (\gamma^0_1,\gamma_2^0) (\Gamma \times \Gamma))\right|}^* \gg M^{-O(1)} |I|.\end{align*} $$

By shifting $\gamma _1^0, \gamma _2^0$ by elements of $\Gamma $ if necessary we may assume that they lie at distance $O(M^{O(1)})$ from the identity. If we partition I into subintervals J of length $\asymp M^{-C} |I|$ for some large constant C, we see from the pigeonhole principle (and Lemma 2.2(i)) that we can find one such J for which

$$ \begin{align*}{\left|\sum_{n \in J} F \otimes \overline{F}( (\varepsilon_1,\varepsilon_2)(n) (g^{\prime}_1,g^{\prime}_2)(n) (\gamma^0_1,\gamma_2^0) (\Gamma \times \Gamma))\right|}^* \gg M^{-O(1)} |J|.\end{align*} $$

As $(\varepsilon _1,\varepsilon _2)$ is $(M,I)$ -smooth, it fluctuates by $O(M^{1-C})$ on J and stays a distance $O(M)$ from the identity, hence by the Lipschitz nature of $F \otimes \overline {F}$ we conclude (for $C=O(1)$ large enough) that there exists $(\varepsilon _1^0, \varepsilon _2^0) \in G \times G$ at distance $O(M)$ from the identity such that

$$ \begin{align*}{\left|\sum_{n \in J} F \otimes \overline{F}( (\varepsilon^0_1,\varepsilon^0_2) (g^{\prime}_1,g^{\prime}_2)(n) (\gamma^0_1,\gamma_2^0) (\Gamma \times \Gamma))\right|}^* \gg M^{-O(1)} |J|.\end{align*} $$

Allowing implied constants to depend on C, we conclude that

$$ \begin{align*}{\left|\sum_{n \in I} F \otimes \overline{F}( (\varepsilon^0_1,\varepsilon^0_2) (g^{\prime}_1,g^{\prime}_2)(n) (\gamma^0_1,\gamma_2^0) (\Gamma \times \Gamma))\right|}^* \gg M^{-O(1)} |I|.\end{align*} $$

From the Baker–Campbell–Hausdorff formula and the M-rationality of $(\gamma _1^0, \gamma _2^0)$ , we see that $(\gamma ^0_1,\gamma _2^0) (\Gamma \times \Gamma ) (\gamma ^0_1, \gamma _2^0)^{-1}$ can be covered by $O(M^{O(1)})$ cosets of $\Gamma \times \Gamma $ , and conversely. Thus, if we set

then $G' \cap (\Gamma \times \Gamma )$ can be covered by $O(M^{O(1)})$ cosets of $\Gamma "$ , thus $\Gamma "$ is a subgroup of $G' \cap (\Gamma \times \Gamma )$ of index $O(M^{O(1)})$ such that

(2.15) $$ \begin{align} \Gamma" (\gamma^0_1,\gamma_2^0) \subset (\gamma^0_1,\gamma_2^0) (\Gamma \times \Gamma). \end{align} $$

Indeed, one can take $\Gamma "$ to be the intersection of $G' \cap (\Gamma \times \Gamma )$ and $(\gamma ^0_1,\gamma _2^0) (\Gamma \times \Gamma ) (\gamma ^0_1, \gamma _2^0)^{-1}$ . One can then write the above claim as

$$ \begin{align*}{\left|\sum_{n \in I} F'( (g^{\prime}_1,g^{\prime}_2)(n) \Gamma")\right|}^* \gg M^{-O(1)} |I|,\end{align*} $$

where $F' \colon G' / \Gamma " \to \mathbb {C}$ is defined by

for any $(g^{\prime }_1,g^{\prime }_2) \in G'$ , with the inclusion (2.15) ensuring that this function is well-defined. Since F is Lipschitz with norm $1/\delta \leq M$ , and $\varepsilon ^0_1, \gamma ^0_1, \varepsilon ^0_2, \gamma ^0_2$ are at distance $O(M^{O(1)})$ from the identity, this function is Lipschitz with norm $O(M^{O(1)})$ , hence by total equidistribution of $(g^{\prime }_1,g^{\prime }_2)$ we conclude (for A large enough) that

(2.16) $$ \begin{align} \left|\int_{G'/\Gamma"} F'\right| \gg M^{-O(1)}. \end{align} $$

Suppose that the slice

is nontrivial. This is a nontrivial closed connected subgroup of G; by considering the final nontrivial element of the series H, $[H,G]$ , $[[H,G],G]$ , $\dots $ , we conclude that H contains a nontrivial closed connected central subgroup of G. Since $Z(G)$ is one-dimensional, we conclude that H contains $Z(G)$ . In particular, $G'$ contains $Z(G) \times \{1\}$ .

Since F has central frequency $\xi $ , we see that

$$ \begin{align*}F'((z,1)(g_1,g_2)) = e(\xi \cdot z) F'(g_1,g_2)\end{align*} $$

for all $z \in Z(G)$ . By invariance of Haar measure, this implies that

$$ \begin{align*}\int_{G'/\Gamma"} F' = e(\xi \cdot z) \int_{G'/\Gamma"} F'.\end{align*} $$

Since $\xi $ is nontrivial, this implies that $\int _{G'/\Gamma "} F' =0$ , contradicting equation (2.16). Thus, the slice $\{ g \in G: (g,1) \in G'\}$ is trivial. Similarly, the slice $\{ g \in G: (1,g) \in G' \}$ is trivial.

Now, suppose that the projection is not all of G. This is a proper closed connected subgroup of G with

$$ \begin{align*}\log K = \{ X \in \log G: (X,Y) \in \log G' \text{ for some } Y \in \log G \};\end{align*} $$

thus $\log K$ is the projection of $\log G'$ to $\log G$ . Since $\log G'$ is $M^{O(1)}$ -rational, $\log K$ is also. Hence, there exists a nontrivial horizontal character $\eta \colon G \to \mathbb {R}/\mathbb {Z}$ of Lipschitz norm $O(M^{O(1)})$ that annihilates K, so in particular $\eta (g^{\prime }_1(n)) = 0$ for all n. From equation (2.14), we then have

$$ \begin{align*}\eta(g_1(n)) = \eta(\varepsilon_1(n)) + \eta(\gamma_1(n)).\end{align*} $$

Since $\gamma _1$ is M-rational, $M\eta (\gamma _1(n)) = 0$ . Thus, if we replace $\eta $ by $M\eta $ we have

$$ \begin{align*}\eta(g_1(n)) = \eta(\varepsilon_1(n)).\end{align*} $$

Since $(\varepsilon _1,\varepsilon _2)$ is $(M,I)$ smooth, we thus conclude that

$$ \begin{align*}\| \eta \circ g_1 \|_{C^\infty(I)} \ll M^{O(1)}\end{align*} $$

and we are in conclusion (i) of the proposition. Thus, we may assume that the projection $\{ g_1 \in G: (g_1,g_2) \in G' \text { for some } g_2 \in G \}$ is all of G. Similarly, we may assume that $\{ g_2 \in G: (g_1,g_2) \in G' \text { for some } g_1 \in G \}$ is all of G.

Applying Goursat’s lemma, we now conclude that $G'$ takes the form

$$ \begin{align*}G' = \{ (g_1, \phi(g_1)): g_1 \in G \}\end{align*} $$

for some group automorphism $\phi \colon G \to G$ . Since $G'$ is a $O(M^{O(1)})$ -rational subgroup of $G \times G$ , $\phi $ must be a Lie group automorphism whose associated Lie algebra automorphism $\log \phi \colon \log G \to \log G$ has coefficients that are rational of height $O(M^{O(1)})$ in the Mal’cev basis. Since $(g^{\prime }_1(n),g^{\prime }_2(n))$ takes values in $G'$ , we have

$$ \begin{align*}g^{\prime}_2(n) = \phi(g^{\prime}_1(n))\end{align*} $$

and hence by equation (2.14) and some rearranging

$$ \begin{align*}g_2(n) = \varepsilon_2(n) \phi(\varepsilon_1(n))^{-1} \phi(g_1(n)) \phi(\gamma_1(n))^{-1} \gamma_2(n).\end{align*} $$

It is then routine to verify that conclusion (ii) of the proposition holds.

Proof. We allow implied constants to depend on $d,D$ . From equation (2.17), one can find progressions $P_i \subset I$ for $i=1,\dots ,K$ such that

$$ \begin{align*}\left|\sum_{n \in I} f(n) 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right| \geq \delta |I|\end{align*} $$

and thus

$$ \begin{align*}\left|\sum_{i=1}^K \theta_i \sum_{n \in I} f(n) 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right| \geq \delta K |I|\end{align*} $$

for some complex numbers $\theta _i$ with $|\theta _i| \leq 1$ . By interchanging the sums and applying Cauchy–Schwarz, we have

$$ \begin{align*}\left|\sum_{i=1}^K \theta_i \sum_{n \in I} f(n) 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right|{}^2 \leq \frac{1}{\delta} |I| \sum_{n \in I} \left| \sum_{i=1}^K \theta_i 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right|{}^2\end{align*} $$

and thus

$$ \begin{align*}\sum_{n \in I} \left| \sum_{i=1}^K \theta_i 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right|{}^2 \geq \delta^{3} K^2 |I|.\end{align*} $$

From the triangle inequality, we have

$$ \begin{align*}\sum_{n \in I} \left|\sum_{i=1}^K \theta_i 1_{P_i}(n) \overline{F}(g_i(n) \Gamma)\right|{}^2 \leq \sum_{1 \leq i,j \leq K} {\left|\sum_{n \in I} F(g_i(n) \Gamma) \overline{F}(g_j(n) \Gamma)\right|}^*\end{align*} $$

and thus

$$ \begin{align*}\sum_{1 \leq i,j \leq K} {\left|\sum_{n \in I} F(g_i(n) \Gamma) \overline{F}(g_j(n) \Gamma)\right|}^* \geq \delta^{3} K^2 |I|.\end{align*} $$

The inner sum is $O(\delta ^{-2} |I|)$ , thus we have

$$ \begin{align*}{\left|\sum_{n \in I} F(g_i(n) \Gamma) \overline{F}(g_j(n) \Gamma)\right|}^* \gg \delta^{O(1)} |I|\end{align*} $$

for $\gg \delta ^{O(1)} K^2$ pairs $(i,j) \in \{1,\dots ,K\}^2$ . For each such pair, we apply Proposition 2.14. If conclusion (i) of that proposition holds for $\gg \delta ^{O(1)} K^2$ pairs $(i,j)$ , then by the pigeonhole principle (noting that there are only $O(\delta ^{-O(1)})$ choices for $\eta $ ) we obtain conclusion (i) of the current proposition. Thus, we may assume that conclusion (ii) of Proposition 2.14 holds for $\gg \delta ^{O(1)} K^2$ pairs $(i,j) \in \{1,\dots ,K\}^2$ , thus we have

$$ \begin{align*}g_i = \varepsilon_{ij} \phi_{ij}(g_j) \gamma_{ij}\end{align*} $$

for all such pairs $(i,j)$ , where $\varepsilon _{ij}$ is $(O(\delta ^{-O(1)}),I)$ -smooth, $\gamma _{ij}$ is $O(\delta ^{-O(1)})$ -rational and $\phi _{ij} \colon G \to G$ is a Lie group automorphism whose associated Lie algebra isomorphism $\log \phi \colon \log G \to \log G$ has matrix coefficients that are all rational of height $O(\delta ^{-O(1)})$ in the Mal’cev basis $X_1,\dots ,X_D$ of $\log G$ . The total number of choices for $\phi _{ij}$ is $O(\delta ^{-O(1)})$ , so by the pigeonhole principle we may assume that $\phi _{ij} = \phi $ is independent of $i,j$ . By Cauchy–Schwarz, we may thus find $\gg \delta ^{O(1)} K^3$ triples $(i,i',j) \in \{1,\dots ,K\}^3$ such that

$$ \begin{align*}g_i = \varepsilon_{ij} \phi(g_j) \gamma_{ij}; \quad g_{i'} = \varepsilon_{i'j} \phi(g_j) \gamma_{i'j},\end{align*} $$

where $\varepsilon _{ij}, \varepsilon _{i'j}, \gamma _{ij}, \gamma _{i'j}$ are as above. This implies that

$$ \begin{align*}g_i = \varepsilon_{ij} \varepsilon_{i'j}^{-1} g_{i'} \gamma_{i'j}^{-1} \gamma_{ij}.\end{align*} $$

Pigeonholing in j and relabeling $i,i'$ as $i,j$ , we obtain conclusion (ii) of the current proposition.

Proof. Using Heath-Brown’s identity (see [Reference Iwaniec and Kowalski37, (13.37), (13.38)] with $K = L$ and $z = (2X)^{1/L}$ ), we have

$$ \begin{align*}\Lambda(n) = \sum_{1 \leq j \leq L} (-1)^{j-1} \binom{L}{j} \sum_{m_1,\ldots, m_j \leq (2X)^{1/L}} \mu(m_1) \cdots \mu(m_j) \sum_{m_1\cdots m_jn_1 \cdots n_j=n} \log n_1\end{align*} $$

and

$$ \begin{align*}\mu(n) = \sum_{1 \leq j \leq L} (-1)^{j-1} \binom{L}{j} \sum_{m_1,\ldots,m_j \leq (2X)^{1/L}} \mu(m_1) \cdots \mu(m_j) \sum_{m_1\cdots m_jn_1\cdots n_{j-1}=n} 1.\end{align*} $$

The conclusion follows after dyadic division of the ranges of variables.

Proof. Write $\ell = mn$ , and note that since $m> D$ , we have $\Omega (\ell ) \geq \frac {\log D}{\log Q}$ . Hence,the left-hand side of equation (2.18) is

$$ \begin{align*} \leq \sum_{\substack{X < \ell \leq X+H \\\Omega(\ell)\geq \frac{\log D}{\log Q}}} d_2(\ell) d_k(\ell)^r \leq e^{-C\frac{\log D}{\log Q}}\sum_{X< \ell \leq X+H} e^{C\Omega(\ell)} d_2(\ell) d_k(\ell)^r \ll_C H\frac{(\log X)^{2k^r e^C}}{\exp(C\frac{\log D}{\log Q})} \end{align*} $$