Proof. The indicator function of $V$ can be written as

\[ \frac{1}{p^d}\sum_{\xi\in V^\perp}e_p(\xi\cdot x), \]

so we have that

\[ |\mathbb{E}_{x}f(x-w)V(x)|\leq\frac{1}{p^d}\sum_{\xi\in V^\perp} |\mathbb{E}_{x}f(x)e_p(\xi\cdot x)|\leq \|f\|_{U^2(\mathbb{F}_p^n)}, \]

by the Gowers–Cauchy–Schwarz inequality. Since $|V|=|\mathbb {F}_p^n|/p^d$, the desired result follows.

Proof. Set $\beta :=\prod _{j=1}^d\beta _j$. Note that $\mathbb {E}_{\mathbf {x}\in (\mathbb {F}_p^n)^t}\|F(\mathbf {x},\cdot )-\beta \|_{U^2(\mathbb {F}_p^n)}^4$ equals

\[ \sum_{\omega\in\{0,1\}^4}(-\beta)^{|\omega|}\mathbb{E}_{\mathbf{x}\in(\mathbb{F}_p^n)^t} \mathbb{E}_{y,h,k}f_1^{\omega}(\mathbf{x},y)f_2^{\omega}(\mathbf{x},y+h)f_3^{\omega} (\mathbf{x},y+k)f_4^{\omega}(\mathbf{x},y+h+k), \]

where

\[ f_i^{\omega}(\mathbf{x},y)=\begin{cases} 1 & \omega_i=1 \\ F(\mathbf{x},y) & \omega_i=0 \end{cases} \]

for each $\omega \in \{0,1\}^4$ and $1\leq i\leq 4$. Since (19) has Cauchy–Schwarz complexity at most $s-1$ by hypothesis, Corollary 3.3 implies that

\[ \mathbb{E}_{\mathbf{x}\in(\mathbb{F}_p^n)^t}\mathbb{E}_{y,h,k}f_1^{\omega}(\mathbf{x},y)f_2^{\omega} (\mathbf{x},y+h)f_3^{\omega}(\mathbf{x},y+k)f_4^{\omega}(\mathbf{x},y+h+k)=\beta^{4-|\omega|}+O_{d}(\varepsilon) \]

for every $\omega \in \{0,1\}^4$. Thus,

\[ \mathbb{E}_{\mathbf{x}\in(\mathbb{F}_p^n)^t}\|F(\mathbf{x},\cdot)-\beta\|_{U^2(\mathbb{F}_p^n)}^4\ll_{d}\varepsilon \]

Markov's inequality then gives

\[ \mathbb{P}(\mathbf{x}\in (\mathbb{F}_p^n)^t : \|F(\mathbf{x},\cdot)-\beta\|_{U^2(\mathbb{F}_p^n)}\geq r)\ll_{d}\frac{\varepsilon}{r^4} \]

for every $r>0$. Taking $r=\varepsilon ^{1/8}$ gives the conclusion of the lemma.

Proof. We write

\[ \|A-\alpha\|_{U^s(\mathbb{F}_p^n)}^{2^s}=\mathbb{E}_{x,h_1,\ldots,h_s} \prod_{\omega\in\{0,1\}^s}(A-\alpha)(x+\omega\cdot(h_1,\ldots,h_s)) \]

and then, for each $\omega \in \{0,1\}^s$, insert the identity

\[ (A-\alpha)(x+\omega\cdot(h_1,\ldots,h_s))=\frac{1}{\rho} \mathbb{E}_{k_\omega}(\Phi-\alpha\rho)(x+\omega\cdot(h_1,\ldots,h_s),k_\omega) \]

to get that $\|A-\alpha \|_{U^s(\mathbb {F}_p^n)}^{2^s}$ equals

\[ \frac{1}{\rho^{2^s}}\mathbb{E}_{x,h_1,\ldots,h_s}\mathbb{E}_{\substack{k_\omega\in\mathbb{F}_p^n \\ \omega\in\{0,1\}^s}}\prod_{\omega\in\{0,1\}^s}(\Phi-\alpha\rho)(x+\omega\cdot(h_1,\ldots,h_s),k_\omega). \]

We can average the above quantity over $y,\ell _1,\ldots,\ell _s$ and make the change of variables $k_\omega \mapsto k_\omega +y+\omega \cdot (\ell _1,\ldots,\ell _s)$ to get that it equals

(20)\begin{equation} \frac{1}{\rho^{2^s}}\mathbb{E}_{\substack{k_\omega\in\mathbb{F}_p^n \\ \omega\in\{0,1\}^s}}\mathbb{E}_{x,y,h_1,\ldots,h_s,\ell_1,\ldots,\ell_s} \prod_{\omega\in\{0,1\}^s}(\Phi-\alpha\rho)((x,k_\omega+y)+ \omega\cdot((h_1,\ell_1),\ldots,(h_s,\ell_s))). \end{equation}

Applying the Gowers–Cauchy–Schwarz inequality bounds (20) above by

\[ \frac{1}{\rho^{2^s}}\mathbb{E}_{\substack{k_\omega\in\mathbb{F}_p^n \\ \omega\in\{0,1\}^s}}\prod_{\omega\in\{0,1\}^s}\|(\Phi-\alpha\rho) (\cdot,k_\omega+\cdot)\|_{U^{s}(\mathbb{F}_p^n\times\mathbb{F}_p^n)}= \frac{1}{\rho^{2^s}}\|\Phi-\alpha\rho\|^{2^s}_{U^{s}(\mathbb{F}_p^n\times\mathbb{F}_p^n)}, \]

which gives us the conclusion of the lemma.

Proof. We proceed by induction on $s$, beginning with the case $s=0$. If $\phi$ is a $\delta$-approximate polynomial of degree at most $0$ on $A$, then $\mathbb {E}_{x,y\in A}1_{\phi (x)=\phi (y)}\geq \delta$, so that, by the pigeonhole principle, there exists some $z\in \mathbb {F}_p^n$ for which $\mu _{A}(\{x\in A:\phi (x)=z\})\geq \delta$. Set $X:=\{x\in A:\phi (x)=z\}$, and consider the set of quadruples

\[ X':=\{(x,x+h,x+k,x+h+k)\in A^4:x,x+h,x+k,x+h+k\in X\}. \]

Note that if $(x,x+h,x+k,x+h+k)\in X'$, then

\[ \phi(x)=\phi(x+h)=\phi(x+k)=\phi(x+h+k)=z, \]

so certainly the derivatives

\[ \phi(x)-\phi(x+h),\phi(x)-\phi(x+k),\phi(x+h)-\phi(x+h+k),\quad \text{and}\quad \phi(x+k)-\phi(x+h+k) \]

of $\phi$ on each of the $1$-dimensional faces of the parallelopiped $(x,x+h,x+k,x+h+k)$ vanish. Since

\[ \alpha\delta\leq\|X\|_{U^1(\mathbb{F}_p^n)}\leq\|X\|_{U^2(\mathbb{F}_p^n)}=\bigg(\frac{|X'|}{p^{3n}}\bigg)^{1/4}, \]

we must have $|X'|\geq (\alpha \delta )^4p^{3n}$. Taking $c_0=8$, the total number of quadruples $(x,x+h,x+k,x+h+k)$ in $A^4$ is $(\alpha ^4+O(C_0\alpha ^8))p^{3n}$ by Corollary 3.3, which means that $X'$ consists of at least a $\delta ^4/2$-proportion of parallelograms $(x,x+h,x+k,x+h+k)$ in $A^4$ if $C_0$ is chosen small enough. Thus, for at least a $\delta ^4/2$-proportion of parallelograms $(x,x+h,x+k,x+h+k)$ in $A^4$, the derivative of $\phi$ on each $1$-dimensional face vanishes, as desired.

Now suppose that the result holds for a general degree $s-1\geq 0$, and let $\phi$ be a $\delta$-approximate polynomial of degree at most $s$ on $A$. By Corollary 3.3,

\[ \mathbb{E}_{h\in\mathbb{F}_p^n}\|\Delta_{h}A-\alpha^2\|_{U^{2s}(\mathbb{F}_p^n)}^{2^{2s}}\ll_s C_s(\alpha\delta)^{c_s}, \]

so that, as long as $C_s$ is sufficiently small and $c_s$ is sufficiently large, it follows from Markov's inequality that

\[ \|\Delta_hA-\alpha^2\|_{U^{2s}(\mathbb{F}_p^n)}<\frac{C_{s-1}(\alpha\delta/2)^{2c_{s-1}}}{2}, \]

and, thus,

\[ |\mu_{\mathbb{F}_p^n}(A\cap (A-h))-\alpha^2|<\frac{C_{s-1}(\alpha\delta/2)^{2c_{s-1}}}{2} \]

as well, for all but a $O(\delta ^2)$-proportion of $h\in \mathbb {F}_p^n$. Thus, for at least a $\Omega (\delta )$-proportion of $h_{s+1}\in \mathbb {F}_p^n$, we have $\|\Delta _{h_{s+1}}A-\alpha ^2\|_{U^{2s}(\mathbb {F}_p^n)}< C_{s-1}(\alpha \delta /2)^{2c_{s-1}}/2$, $|\mu _{\mathbb {F}_p^n}(A\cap (A-h_{s+1}))-\alpha ^2|< C_{s-1}(\alpha \delta /2)^{2c_{s-1}}/2$, and that the function $\partial _{h_{s+1}}\phi$ is a $O(\delta )$-approximate polynomial of degree at most $s-1$ on $A\cap (A-h_{s+1})$. Denoting the set of such $h_{s+1}$ by $H$, so that $\mu _{\mathbb {F}_p^n}(H)\gg \delta$, the induction hypothesis then says that, for each $h\in H$, there are at least a $\Omega _{s}(\delta ^{O_{s}(1)})$-proportion of $2s$-dimensional parallelopipeds $(x+\omega \cdot (h_1,\ldots,h_{2s}))_{\omega \in \{0,1\}^{2s}}$ in $(A\cap (A-h))^{2^{2s}}$ for which the derivative of $\partial _h\phi$ on each $(2s-1)$-dimensional face vanishes.

Summing over all $h\in H$, it follows that, for at least a $\Omega _{s}(\delta ^{O_s(1)})$-proportion of $(2s+2)$-tuples $(x,y,h_1,\ldots,h_{2s})$ for which $(x+\omega \cdot (h_1,\ldots,h_{2s}))_{\omega \in \{0,1\}^{2s}}$ and $(y+\omega \cdot (h_1,\ldots,h_{2s}))_{\omega \in \{0,1\}^{2s}}$ are both in $A^{2^{2s}}$, one has

\[ \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(x+\epsilon h_i)= \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(y+\epsilon h_i) \]

for all $i=1,\ldots,2s$ and $\epsilon =0,1$. By Corollary 3.3,

\[ \mathbb{E}_{h_1,\ldots,h_{2s}}\|\Delta_{h_1,\ldots,h_{2s}}A-\alpha^{2^{2s}}\|_{U^{1} (\mathbb{F}_p^n)}^2\ll_sC_s(\alpha\delta)^{c_s}, \]

and so, by Markov's inequality,

(21)\begin{equation} \mu_{\mathbb{F}_p^n}\bigg(\bigcap_{\omega\in\{0,1\}^{2s}} [A-\omega\cdot(h_1,\ldots,h_{2s})]\bigg)=\alpha^{2^{2s}}+O([C_s(\alpha\delta)^{c_s}]^{\Omega(1)}) \end{equation}

for all but a $O([C_s(\alpha \delta )^{c_{s}}]^{\Omega (1)})$-proportion of $(h_1,\ldots,h_{2s})$ in $(\mathbb {F}_p^n)^{2s}$. By taking $C_s$ small enough and $c_s$ large enough, there are therefore at least a $\Omega _{s}(\delta ^{O_s(1)})$-proportion of $2s$-tuples $(h_1,\ldots,h_{2s})$ in $(\mathbb {F}_p^n)^{2s}$ for which (21) holds and, for at least a $\Omega _{s}(\delta ^{O_s(1)})$-proportion of pairs $(x,y)\in A^2$ such that $(x+\omega \cdot (h_1,\ldots,h_{2s}))_{\omega \in \{0,1\}^{2s}}$ and $(y+\omega \cdot (h_1,\ldots,h_{2s}))_{\omega \in \{0,1\}^{2s}}$ are both in $A^{2^{2s}}$, one also has

\[ \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(x+\epsilon h_i)= \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(y+\epsilon h_i) \]

for all $i=1,\ldots,2s$ and $\epsilon =0,1$. For each such $2s$-tuple $\mathbf {h}$, it follows from the pigeonhole principle that there exists a $y_{\mathbf {h}}$ in the set

\[ A_{\mathbf{h}}:=\bigcap_{\omega\in\{0,1\}^{2s}}(A-\omega\cdot(h_1,\ldots,h_{2s})) \]

such that, for at least a $\Omega _{s}(\delta ^{O_s(1)})$-proportion of $x\in A_{\mathbf {h}}$, one has

\[ \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(x+\epsilon h_i)= \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}}\phi(y_{\mathbf{h}}+\epsilon h_i) \]

for all $i=1,\ldots,2s$ and $\epsilon =0,1$.

Now set $v_{i,\epsilon,\mathbf {h}}:=\partial _{h_1,\ldots,\widehat {h_i},\ldots,h_{2s}}\phi (y_{\mathbf {h}}+\epsilon h_i)$,

\[ X_{\mathbf{h}}:=\{x\in A_{\mathbf{h}}:\partial_{h_1,\ldots, \widehat{h_i},\ldots,h_{2s}}\phi(x+\epsilon h_i)=v_{i,\epsilon,\mathbf{h}}\ \text{for all}\ i=1,\ldots,2s\text{ and }\epsilon=0,1\}, \]

so that $\mu _{A_{\mathbf {h}}}(X_{\mathbf {h}})\gg _s\delta ^{O_s(1)}$, and

\[ X_{\mathbf{h}}':=\{(x,x+k,x+k',x+k+k')\in A_{\mathbf{h}}^4:x,x+k,x+k',x+k+k'\in X_{\mathbf{h}}\}. \]

Note that $(x,x+k,x+k',x+k+k')\in X'_{\mathbf {h}}$ if and only if

\[ \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s}} \phi(x+\epsilon h_i+\omega'\cdot(k,k'))=v_{i,\epsilon,\mathbf{h}} \]

for all $i=1,\ldots,2s$, $\epsilon =0,1$, and $\omega '\in \{0,1\}^2$. Thus, whenever $(x,x+k,x+k',x+k+k')\in X'_{\mathbf {h}}$, we have

\begin{align*} \partial_{h_1,\ldots,h_{2s},k}\phi(x)=\partial_{h_1,\ldots,h_{2s},k'} \phi(x)=\partial_{h_1,\ldots,h_{2s},k}\phi(x+k')&=\partial_{h_1,\ldots,h_{2s},k'}\phi(x+k) \\ &=v_{1,0,\mathbf{h}}-v_{1,1,\mathbf{h}}-(v_{1,0,\mathbf{h}}-v_{1,1,\mathbf{h}}) \\ &=0 \end{align*}

and

\[ \partial_{h_1,\ldots,\widehat{h_i},\ldots,h_{2s},k,k'} \phi(x+\epsilon h_i)=v_{i,\varepsilon,\mathbf{h}}-v_{i,\varepsilon,\mathbf{h}}-v_{i,\varepsilon,\mathbf{h}}+v_{i,\varepsilon,\mathbf{h}}=0 \]

for all $i=1,\ldots,2s$ and $\epsilon =0,1$. That is, the derivative of $\phi$ vanishes on all $(2s+1)$-dimensional faces of the $(2s+2)$-dimensional parallelopiped $(x+\omega \cdot (h_1,\ldots,h_{2s},k,k'))_{\omega \in \{0,1\}^{2s+2}}$.

As in the $s=0$ case,

\[ \delta^{O_s(1)}\mu_{\mathbb{F}_p^n}(A_{\mathbf{h}})\ll_s\|X_{\mathbf{h}}\|_{U^1(\mathbb{F}_p^n)}\leq \|X_{\mathbf{h}}\|_{U^2(\mathbb{F}_p^n)}=\bigg(\frac{\# X'_{\mathbf{h}}}{p^{3n}}\bigg)^{1/4}, \]

so that

\[ \# X'_{\mathbf{h}}\gg_s \delta^{O_s(1)}\mu_{\mathbb{F}_{p}^n}(A_{\mathbf{h}})^4p^{3n} \]

for at least a $\Omega _s(\delta ^{O_s(1)})$-proportion of $2s$-tuples $\mathbf {h}$. Each ordered quadruple $(x,\mathbf {h},k,k')$ for which $(x,x+k,x+k',x+k+k')\in X_{\mathbf {h}}'$ corresponds to a unique $(2s+2)$-dimensional parallelopiped

\[ P(x,\mathbf{h},k,k')=(x+\omega\cdot(h_1,\ldots,h_{2s},k,k'))_{\omega\in\{0,1\}^{2s+2}} \]

in $A^{2s+2}$. Thus,

\begin{align*} &\#\{P(x,\mathbf{h},k,k'):\mathbf{h}\in(\mathbb{F}_p^n)^{2s}\text{ and } (x,x+k,x+k',x+k+k')\in X_{\mathbf{h}}'\} \\ &\quad \gg_s\delta^{O_s(1)}(\alpha^{2^{2s+2}}+O([C_s(\alpha\delta)^{c_s}]^{\Omega(1)}))p^{(2s+3)n}. \end{align*}

In comparison, the number of $(2s+2)$-dimensional parallelopipeds in $A$ is

\[ (\alpha^{2^{2s+2}}+O(C_s(\alpha\delta)^{c_s}))p^{(2s+3)n} \]

by Corollary 3.3. The conclusion of the lemma now follows as long as $C_s$ is sufficiently small and $c_s$ is sufficiently large.

Proof of Lemma 4.1 We proceed by induction on $r$ and $s$, beginning with the $r=1$, $s=0$ caseFootnote ¹. Since $\operatorname {codim}\{y\in \mathbb {F}_p^n:\Phi (x,y)=1\}=d$ for all $x\in A$, certainly

\[ \operatorname{codim}\{y\in\mathbb{F}_p^n:\Phi(\psi_1(\mathbf{x}),y+w_1)=1\}= \operatorname{codim}(\{y\in\mathbb{F}_p^n:\Phi(\psi_1(\mathbf{x}),y)=1\}-w_1)=d \]

for all $\mathbf {x}\in (\mathbb {F}_p^n)^m$ for which $\psi _1(\mathbf {x})\in A$, and this case follows trivially without even needing the assumption that $\|\Phi -\alpha \rho \|_{U^2(\mathbb {F}_p^n\times \mathbb {F}_p^n)}$ is small.

Now let $r\geq 2$ or $s\geq 1$, and assume that the result holds for all pairs of integers $(r',s')$ satisfying:

1. (i) $0\leq r'< r$ and $1\leq s'\leq s$; or

2. (ii) $1\leq s'< s$;

and let $C'$ be at most the minimum of $C_{r',s'}$ and $c'$ be at least the maximum of $c_{r',s'}$ over all such pairs with $r'<\max (r,2^{s+1})$. As long as $\|\Phi -\alpha \rho \|_{U^{2s+2}(\mathbb {F}_p^n\times \mathbb {F}_p^n)}< C'(\alpha \delta \rho ^{2r}/2)^{c'2^r}$, it follows from the induction hypothesis that for all but a $O(\delta )$-proportion of $\mathbf {x}\in (\mathbb {F}_p^n)^m$ for which $\psi _1(\mathbf {x}),\ldots,\psi _r(\mathbf {x})\in A$, we must have

\[ \mathbb{E}_{y}\prod_{i=1}^r(\Phi-\rho)(\psi_i(\mathbf{x}),y+w_i)= \mathbb{E}_{y}\prod_{i=1}^r\Phi(\psi_i(\mathbf{x}),y+w_i)-\rho^r \]

for all $w_1,\ldots,w_r\in \mathbb {F}_p^n$. If the codimension of some $\{y\in \mathbb {F}_p^n:\prod _{i=1}^{r}\Phi (\psi _i(\mathbf {x}),y+w_i)=1\}$ is not $rd$ for one of these typical $\mathbf {x}$, then it is either $n$ or at most $rd-1$, which means that

(22)\begin{equation} \bigg|\mathbb{E}_{y}\prod_{i=1}^r(\Phi-\rho)(\psi_i(\mathbf{x}),y+w_i)\bigg|\geq\frac{\rho^r}{2} \end{equation}

in either case, since $p\geq 2$. Squaring both sides of (22), multiplying by $\prod _{i=1}^rA(\psi _i(\mathbf {x}))$, averaging over all $\mathbf {x}\in (\mathbb {F}_p^n)^m$, swapping the order of summation, and applying Lemma 4.2 (to deduce the uniformity of $A$) and Theorem 3.2 yields

\[ \mathbb{E}_{y,z}\|(\Phi-\rho A)(\cdot,y)(\Phi-\rho A)(\cdot,z)\|_{U^{s+1}(\mathbb{F}_p^n)}\gg \delta\rho^{2r}(\alpha+O_{s,r}((C')^{\Omega_s(1)}\alpha^{c'-1})). \]

By Hölder's inequality, we then have

\[ \mathbb{E}_{x,h_1,\ldots,h_{s+1}}|\mathbb{E}_{y}\Delta_{(h_1,0),\ldots,(h_{s+1},0)}(\Phi-\rho A)(x,y)|^2\gg_s (\delta\rho^{2r})^{2^{s+1}}(\alpha^{2^{s+1}}+O_{s,r}((C')^{\Omega_s(1)}\alpha^{c'-1})). \]

It follows from this, the induction hypothesis, our assumption that $\|\Phi -\alpha \rho \|_{U^{2s+2}(\mathbb {F}_p^n\times \mathbb {F}_p^n)}< C'(\alpha \delta \rho ^{2r}/2)^{c'2^r}$, and Lemma 4.2 that

\[ \mathbb{E}_{x,h_1,\ldots,h_{s+1}}|\mathbb{E}_{y}\Delta_{(h_1,0),\ldots,(h_{s+1},0)} \Phi(x,y)-\rho^{2^{s+1}}A(x)|^2\gg_s(\delta\rho^{2r})^{2^{s+1}} (\alpha^{2^{s+1}}+O_{s,r}((C')^{\Omega_s(1)}\alpha^{c'-1})), \]

since the Cauchy–Schwarz complexity of any proper subset of

\[ \{x+\omega\cdot(h_1,\ldots,h_{s+1}):\omega\in\{0,1\}^{s+1}\} \]

is at most $s-1$.

Thus, by taking $C'$ sufficiently small and $c'$ sufficiently large, we get that

(23)\begin{equation} \operatorname{codim}\{y\in\mathbb{F}_p^n:\Delta_{(h_1,0),\ldots,(h_{s+1},0)}\Phi(x,y)=1\}\neq 2^{s+1}d \end{equation}

for at least an $\Omega _s((\delta \rho )^{O_s(1)})$-proportion of $(s+2)$-tuples $(x,h_1,\ldots,h_{s+1})\in (\mathbb {F}_p^n)^{s+2}$ for which $(x+\omega \cdot (h_1,\ldots,h_{s+1}))_{\omega \in \{0,1\}^{s+1}}\in A^{2^{s+1}}$. The condition (23) implies that, for each such $(x,h_1,\ldots,h_{s+1})$, there exist vectors $v_{x,\mathbf {h},\omega }\in V^{\perp }_{x+\omega \cdot \mathbf {h}}$, $\omega \in \{0,1\}^{s+1}$, not all of which are zero, such that

\[ \sum_{\omega\in\{0,1\}^{s+1}}v_{x,\mathbf{h},\omega}=0. \]

Fix a basis $\{\gamma _{z,1},\ldots,\gamma _{z,d}\}$ of $V_z^\perp$ for each $z\in \mathbb {F}_p^n$. We can write every vector $v_{x,\mathbf {h},\omega }$ in terms of this basis, giving us that

\begin{align*} \sum_{\omega\in\{0,1\}^{s+1}}\sum_{j=1}^db_{x,\mathbf{h},\omega,j}\gamma_{x+\omega\cdot\mathbf{h},j}=0 \end{align*}

for some $2^{s+1}d$-tuple of constants $(b_{x,\mathbf {h},\omega,j})_{\omega \in \{0,1\}^{s+1},\ 1\leq j\leq d}$, not all of which are zero.

We apply the pigeonhole principle to deduce that there is some $2^{s+1}d$-tuple of constants $(b_{\omega,j})_{\omega \in \{0,1\}^{s+1},\ 1\leq j\leq d}$, not all of which are zero, such that

\[ \sum_{\omega\in\{0,1\}^{s+1}}\sum_{j=1}^db_{\omega,j}\gamma_{x+\omega\cdot\mathbf{h},j}=0 \]

for at least a $\Omega _s((\delta \rho )^{O_s(1)})$-proportion of $(s+2)$-tuples $(x,h_1,\ldots,h_{s+1})\in (\mathbb {F}_p^n)^{s+2}$ for which $(x+\omega \cdot (h_1,\ldots,h_{s+1}))_{\omega \in \{0,1\}^{s+1}}\in A^{2^{s+1}}$. Defining

\[ \phi_{\omega}(z):=\sum_{j=1}^db_{\omega,j}\gamma_{z,j} \]

for each $\omega \in \{0,1\}^{s+1}$, the above says that, for at least a $\Omega _s((\delta \rho )^{O_s(1)})$-proportion of $(s+2)$-tuples $(x,h_1,\ldots,h_{s+1})\in (\mathbb {F}_p^n)^{s+2}$ for which $(x+\omega \cdot (h_1,\ldots,h_{s+1}))_{\omega \in \{0,1\}^{s+1}}\in A^{2^{s+1}}$, we must have

\[ \sum_{\omega\in\{0,1\}^{s+1}}\phi_\omega(x+\omega\cdot\mathbf{h})=0, \]

where at least one of the functions $\phi _{\omega }:\mathbb {F}_p^n\to \mathbb {F}_p^n$ does not have the zero vector in its image (because $\phi _\omega (z)$ is always a nontrivial linear combination of linearly independent vectors). Let $\phi$ be any such $\phi _\omega$. It then follows by applying Corollary 3.3 to the inside average of

\[ \mathbb{E}_{y}\mathbb{E}_{x,h_1,\ldots,h_{s+1}}e_p \bigg(y\cdot\sum_{\omega\in\{0,1\}^{s+1}} \phi_\omega(x+\omega\cdot \mathbf{h})\bigg)A(\phi_\omega(x+\omega\cdot \mathbf{h})) \]

that $\partial _{h_1,\ldots,h_{s+1}}\phi (x)=0$ for at least a $\Omega _s((\delta \rho )^{O_s(1)})$-proportion of $(s+2)$-tuples $(x,h_1,\ldots,h_{s+1})\in (\mathbb {F}_p^n)^{s+2}$ for which $(x+\omega \cdot (h_1,\ldots,h_{s+1}))_{\omega \in \{0,1\}^{s+1}}\in A^{2^{s+1}}$, again provided that $C'$ is sufficiently small and $c'$ is sufficiently large. That is, $\phi$ is a $\Omega _s((\delta \rho )^{O_s(1)})$-approximate polynomial of degree at most $s$ on $A$.

If $C'$ is small enough and $c'$ is large enough, Lemmas 4.2 and 4.4, then imply that for at least a $\Omega _{s}((\alpha \delta \rho )^{O_s(1)})$-proportion of $(2s+2)$-dimensional parallelopipeds $(x+\omega \cdot (h_1,\ldots,h_{2s+2}))_{\omega \in \{0,1\}^{2s+2}}$ in $A^{2^{2s+2}}$, the derivative of $\phi$ on each $(2s+1)$-dimensional face vanishes, i.e.

\[ \sum_{\substack{\omega\in\{0,1\}^{2s+2} \\ \omega_i=\epsilon}}(-1)^{|\omega|}\phi(x+\omega\cdot(h_1,\ldots,h_{2s+2}))=0 \]

for all $i=1,\ldots,2s+2$ and $\epsilon =0,1$. Call the set of $(2s+3)$-tuples $(x,h_1,\ldots,h_{2s+2})$ corresponding to such $(2s+2)$-dimensional parallelopipeds $X$. Consider $\|\Phi -\rho A\|_{U^{2s+2}(\mathbb {F}_p^n\times \mathbb {F}_p^n)}^{2^{2s+2}}$, which, plugging in the expression

\[ \rho A(x)\sum_{0\neq v\in V_x^\perp}e_p(v\cdot (y-u)) \]

for $\Phi -\rho A$, equals

\[ \rho^{2^{2s+2}}\mathbb{E}_{x,h_1,\ldots,h_{2s+2}}\Delta_{h_1,\ldots,h_{2s+2}}A(x) \sum_{\substack{0\neq v_\omega\in V_{x+\omega\cdot\mathbf{h}}^\perp \\ \omega\in\{0,1\}^{2s+2}}}\prod_{\substack{1\leq i\leq 2s+2 \\ \epsilon=0,1}}1_{\mathbf{0}}\bigg(\sum_{\substack{\omega\in\{0,1\}^{2s+2} \\ \omega_i=\epsilon}}(-1)^{|\omega|}v_\omega\bigg). \]

The above has size $\gg _s(p-1)(\alpha \rho )^{2^{2s+2}}(\alpha \delta \rho )^{O_s(1)}\gg _s(\alpha \delta \rho )^{O_s(1)}$, coming from the contribution of $v_\omega =\lambda \phi (x+\omega \cdot (h_1,\ldots,h_{2s+2}))$ for each $\lambda \in \mathbb {F}_p^\times$ and $(x,h_1,\ldots,h_{2s+2})\in X$. On the other hand, we have

\begin{align*} \|\Phi-\rho A\|_{U^{2s+2}(\mathbb{F}_p^n\times\mathbb{F}_p^n)} &= \|\Phi-\rho\alpha+ \rho\alpha-\rho A\|_{U^{2s+2}(\mathbb{F}_p^n\times\mathbb{F}_p^n)}\\ &\leq \|\Phi-\rho\alpha\|_{U^{2s+2}(\mathbb{F}_p^n\times\mathbb{F}_p^n)}+\rho\|A-\alpha\|_{U^{2s+2}(\mathbb{F}_p^n)}, \end{align*}

so that

\[ (\alpha\delta\rho)^{O_s(1)}\ll_s \|\Phi-\rho A\|_{U^{2s+2}(\mathbb{F}_p^n\times\mathbb{F}_p^n)}< 2C_{s,r}(\alpha\delta\rho)^{c_{s,r}}. \]

Taking $C_{s,r}$ sufficiently small and $c_{s,r}$ sufficiently large will thus yield a contradiction if

\[ \operatorname{codim}\bigg\{y\in\mathbb{F}_p^n:\prod_{i=1}^r\Phi(\psi_i(\mathbf{x}),y+w_i)=1\bigg\}\neq rd \]

for some $w_1,\ldots,w_r\in \mathbb {F}_p^n$ for a $\delta$-proportion of $\mathbf {x}\in (\mathbb {F}_p^n)^m$ for which $\psi _1(\mathbf {x}),\ldots,\psi _r(\mathbf {x})\in A$.

Proof. The density of $T$ in $\mathbb {F}_p^n\times \mathbb {F}_p^n$ can be written as

\[ \mathbb{E}_{x,y}F(x,y)\Phi(x,y), \]

where $F(x,y):=B(y)C(x+y)D(2x+y)$. Set $L(x,y):=\{y,x+y,2x+y\}$. Lemma 3.6 says that

\[ \mathbb{P}(x\in \mathbb{F}_p^n:\|F(x,\cdot)-\beta\gamma\delta\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon}, \]

since

\[ L(x,y) \cup L(x,y+h) \cup L(x,y+k) \cup L(x,y+h+k) \]

has Cauchy–Schwarz complexity at most $3$. Thus, Lemma 3.5 yields

\[ \mathbb{E}_{y}F(x,y)\Phi(x,y)=(\beta\gamma\delta\rho+O(\varepsilon^{1/8}))A(x) \]

for all but a $O(\sqrt {\varepsilon })$-proportion of $x\in \mathbb {F}_p^n$, so that

\[ \mathbb{E}_{x,y}F(x,y)\Phi(x,y)=\alpha\beta\gamma\delta\rho+O(\varepsilon^{1/8}).\]

Proof of Lemma 2.3 The quantity of interest $\Lambda (T,T,T,S)$ is

\begin{align*} \mathbb{E}_{x,y,z} (& B(y+z)B(y+2z)C(x+y)C(x+y+2z)D(2x+y)D(2x+y+z)\\ & \Phi(x,y)\Phi(x,y+z)S(x+z,y)), \end{align*}

which, after a change of variables, can be written as

\[ \mathbb{E}_{x,y}S(x,y)\mu(x,y), \]

where $\mu (x,y)$ equals

\begin{align*} \mathbb{E}_{z}( & B(y+z)B(y+2z)C(x+y-z)C(x+y+z)D(2x+y-2z)D(2x+y-z)\\ & \Phi(x-z,y)\Phi(x-z,y+z)). \end{align*}

We will show that $\mu (x,y)$ is very close to the constant value $\alpha \beta ^2\gamma ^2\delta ^2\rho ^2$ for almost every pair $(x,y)\in \mathbb {F}_p^n\times \mathbb {F}_p^n$, from which it will then follow that $\Lambda (T,T,T,S)$ is close to $\sigma \alpha ^2\beta ^3\gamma ^3\delta ^3\rho ^3$.

The first moment $\mathbb {E}_{x,y}\mu (x,y)$ equals

\[ \mathbb{E}_{x,y,z}B(y+z)B(y+2z)C(x+y)C(x+y+2z)D(2x+y)D(2x+y+z)\Phi(x,y)\Phi(x,y+z). \]

Applying Lemma 3.6 yields

\[ \mathbb{P}((x,y)\in \mathbb{F}_p^n\times\mathbb{F}_p^n:\|F(x,y,\cdot)-\beta^2\gamma\delta\|_{U^2(\mathbb{F}_p^n)}\geq\varepsilon^{1/8}) \ll\sqrt{\varepsilon}, \]

where

\[ F(x,y,z):=B(y+z)B(y+2z)C(x+y+2z)D(2x+y+z). \]

It therefore follows from Lemma 3.5 that

\[ \mathbb{E}_{z}F(x,y,z)\Phi(x,y+z)=(\beta^2\gamma\delta\rho+O(\varepsilon^{1/8}))A(x) \]

for all but a $O(\sqrt {\varepsilon })$-proportion of $(x,y)\in \mathbb {F}_p^n\times \mathbb {F}_p^n$. Thus,

\[ \mathbb{E}_{x,y}\mu(x,y)=\beta^2\gamma\delta\rho\mathbb{E}_{x,y}C(x+y)D(2x+y)\Phi(x,y)+O(\varepsilon^{1/8}). \]

By arguing as in the proof of Lemma 5.1, we have

\[ \mathbb{E}_{x,y}C(x+y)D(2x+y)\Phi(x,y)=\alpha\gamma\delta\rho+O(\varepsilon^{1/8}), \]

and, thus, conclude that

\[ \mathbb{E}_{x,y}\mu(x,y)=\alpha\beta^2\gamma^2\delta^2\rho^2+O(\varepsilon^{1/8}). \]

The second moment $\mathbb {E}_{x,y}\mu (x,y)^2$ equals

\begin{align*} \mathbb{E}_{x,y,z,h} (&\Delta_{h}B(y+z)\Delta_{2h}B(y+2z) \Delta_{-h}C(x+y)\Delta_{h}C(x+y+2z) \\ &\Delta_{-2h}D(2x+y)\Delta_{-h}D(2x+y+z)\Delta_{(-h,0)}\Phi(x,y)\Delta_{(-h,h)}\Phi(x,y+z)). \end{align*}

Applying Lemma 3.6 again yields

\[ \mathbb{P}((x,y,h)\in\mathbb{F}_p^n\times\mathbb{F}_p^n\times\mathbb{F}_p^n:\|G(x,y,h,\cdot)-\beta^4 \gamma^2\delta^2\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon}, \]

where

\[ G(x,y,h,z):=\Delta_{h}B(y+z)\Delta_{2h}B(y+2z)\Delta_{h}C(x+y+2z)\Delta_{-h}D(2x+y+z), \]

and applying Lemma 4.1 yields

\[ \mathbb{P}((x,x-h)\in A\times A:\operatorname{codim}\{z\in\mathbb{F}_p^n:\Delta_{(-h,h)}\Phi(x,y+z)=1\}\neq 2d) \ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}} \]

for all $y\in \mathbb {F}_p^n$. Thus, by Lemma 3.5,

\[ \mathbb{E}_{y}G(x,y,h,z)\Delta_{(-h,h)}\Phi(x,y+z)=\beta^4\gamma^2\delta^2\rho^2+O(\varepsilon^{1/8}) \]

for all but a $O(\varepsilon ^{\Omega (1)}/\rho ^{O(1)})$-proportion of $(x,x+h,y)\in A\times A\times \mathbb {F}_p^n$, so that

\[ \mathbb{E}_{x,y}\mu(x,y)^2=\beta^4\gamma^2\delta^2\rho^2\mathbb{E}_{x,y,h} \Delta_{-h}C(x+y)\Delta_{-2h}D(2x+y)\Delta_{(-h,0)}\Phi(x,y)+O \bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

By Lemmas 3.6 and 4.1 again, we have

\[ \mathbb{P}((x,h)\in\mathbb{F}_p^n\times\mathbb{F}_p^n:\|H(x,h,\cdot)-\gamma^2 \delta^2\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon} \]

and

\[ \mathbb{P}((x,x-h)\in A\times A:\operatorname{codim} \{y\in\mathbb{F}_p^n:\Delta_{(-h,0)}\Phi(x,y)=1\}\neq 2d)\ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \]

where

\[ H(x,h,y):=\Delta_{-h}C(x+y)\Delta_{-2h}D(2x+y), \]

so that

\[ \mathbb{E}_{x,y,h}\Delta_{-h}C(x+y)\Delta_{-2h}D(2x+y)\Delta_{(-h,0)} \Phi(x,y)=\gamma^2\delta^2\rho^2\mathbb{E}_{x,h}\Delta_{-h}A(x)+O \bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg) \]

by Lemma 3.5. Using Corollary 3.3 to estimate $\mathbb {E}_{x,h}\Delta _{-h}A(x)$, we thus conclude that

\[ \mathbb{E}_{x,y}\mu(x,y)^2 = \alpha^2\beta^4\gamma^4\delta^4\rho^4+O \bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

Our estimates for the first and second moments of $\mu$ imply that $\mu$ has variance $\mathbb {E}_{x,y}|\mu (x,y)-\alpha \beta ^2\gamma ^2\delta ^2\rho |^2\ll \varepsilon ^{\Omega (1)}/\rho ^{O(1)}$. It follows that

\begin{align*} \mathbb{E}_{x,y}S(x,y)\mu(x,y) &= \sigma\alpha^2\beta^3\gamma^3\delta^3\rho^3+O (\mathbb{E}_{x,y}|\mu(x,y)-\alpha\beta^2\gamma^2\delta^2\rho^2|) \\ &= \sigma\alpha^2\beta^3\gamma^3\delta^3\rho^3+O ([\mathbb{E}_{x,y}|\mu(x,y)-\alpha\beta^2\gamma^2\delta^2\rho^2|^2]^{1/2}) \\ &= \sigma\alpha^2\beta^3\gamma^3\delta^3\rho^3+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \end{align*}

When $c_1$ is sufficiently small and $c_2$ is sufficiently large, this gives the desired lower bound for $\Lambda (T,T,T,S)$.

Proof of Lemma 2.4 We prove (16), (17), and then (18), proceeding in decreasing order of the number of applications of Cauchy–Schwarz required. For (16), we make the change of variables $z\mapsto z-x-y$ to write $\Lambda (f_0,f_1,f_2,f_3)$ as

\[ \mathbb{E}_{x,y,z}f_0(x,y)f_1(x,z-x)f_2(x,2z-2x-y)f_3(z-y,y), \]

which, by applying the Cauchy–Schwarz inequality in the $x$ and $z$ variables, has modulus squared bounded by

\[ \mu_{\mathbb{F}_p^n\times\mathbb{F}_p^n}(T)\cdot\mathbb{E}_{x,z}B(z-x)D(x+z)|\mathbb{E}_yf_0(x,y)f_2(x,2z-2x-y)f_3(z-y,y)|^2. \]

The first factor equals $\alpha \beta \gamma \delta \rho +O(\varepsilon ^{1/8})$ by Lemma 5.1. Expanding the square and making a change of variables, the second factor equals

\[ \mathbb{E}_{x,y,z,h_1}B(z-x)D(x+z)\Delta_{(0,h_1)}f_0(x,y) \Delta_{(0,-h_1)}f_2(x,2z-2x-y)\Delta_{(-h_1,h_1)}f_3(z-y,y). \]

By another application of the Cauchy–Schwarz inequality in the $y,z,$ and $h_1$ variables, the modulus squared of this is at most

(24)\begin{equation} \mathbb{E}_{y,z,h_1}\Delta_{h_1}B(y)C(z+y)\Delta_{-h_1}D(2z+y)\Delta_{(-h_1,h_1)}\Phi(z,y) \end{equation}

times

\begin{align*} \mathbb{E}_{y,z,h_1} ( & C(z)\Delta_{(-h_1,h_1)}\Phi(z-y,y)|\mathbb{E}_{x}B(z-x)D(x+z)\\ & \Delta_{(0,h_1)}f_0(x,y)\Delta_{(0,-h_1)}f_2(x,2z-2x-y)|^2). \end{align*}

The first factor (24) can be estimated in the same manner as the averages appearing in the proof of Lemma 2.3, and equals $\alpha ^2\beta ^2\gamma \delta ^2\rho ^2+O(\varepsilon ^{\Omega (1)}/\rho ^{O(1)})$. Expanding the square and making a change of variables, we get that the second factor equals

\begin{align*} \mathbb{E}_{x,y,z,h_1,h_2}(&\Delta_{-h_2}B(y-z)C(x+y-z)\Delta_{h_2}D(2x+y-z)\Delta_{(-h_1,h_1)}\Phi(x+z,y-2z) \\ &\Delta_{(0,h_1),(h_2,0)}f_0(x,y-2z)\Delta_{(0,-h_1),(h_2,-2h_2)}f_2(x,y)). \end{align*}

A final application of the Cauchy–Schwarz inequality in the $x,y,h_1,$ and $h_2$ variables bounds the modulus squared of this by

(25)\begin{equation} \mathbb{E}_{x,y,h_1,h_2}\Delta_{-h_1,-2h_2}B(y)\Delta_{-h_1,-h_2}C(x+y) \Delta_{-h_1}D(2x+y)\Delta_{(0,-h_1),(h_2,-2h_2)}\Phi(x,y) \end{equation}

times

\begin{align*} \mathbb{E}_{x,y,h_1,h_2}&\Delta_{-h_1,-2h_2}B(y)\Delta_{-h_1,-h_2}C(x+y)\Delta_{-h_1}D(2x+y) \Delta_{(h_2,-2h_2)}\Phi(x,y)\\ & \cdot|\mathbb{E}_{z}\Delta_{-h_2}B(y-z)C(x+y-z)\Delta_{h_2}D(2x+y-z)\Delta_{(-h_1,h_1)}\Phi(x+z,y-2z)\\ &\Delta_{(0,h_1),(h_2,0)}f_0(x,y-2z)|^2 \end{align*}

By Lemmas 3.6 and 4.1, we have

\begin{gather*} \mathbb{P}((x,y,h_2)\in\mathbb{F}_p^n\times\mathbb{F}_p^n\times\mathbb{F}_p^n:\|F(x,y,h_2,\cdot)-\beta^2\gamma^2 \delta\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon},\\ \mathbb{P}((x,h_2)\in\mathbb{F}_p^n\times\mathbb{F}_p^n:\|G(x,h_2,\cdot)-\beta^2\gamma^2\delta\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8}) \ll\sqrt{\varepsilon},\\ \mathbb{P}((x,x+h_2)\in A\times A:\operatorname{codim} \{h_1\in\mathbb{F}_p^n:\Delta_{(h_2,-2h_2)}\Phi(x,y-h_1)=1\}\neq 2d) \ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}} \end{gather*}

for all $y\in \mathbb {F}_p^n$, and

\[ \mathbb{P}((x,x+h_2)\in A\times A:\operatorname{codim} \{y\in\mathbb{F}_p^n:\Delta_{(h_2,-2h_2)}\Phi(x,y)=1\}\neq 2d) \ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \]

where

\[ F(x,y,h_2,h_1):=\Delta_{-2h_2}B(y-h_1)\Delta_{-h_2}C(x+y-h_1)D(2x+y-h_1) \]

and

\[ G(x,h_2,y):=\Delta_{-2h_2}B(y)\Delta_{-h_2}C(x+y)D(2x+y). \]

It then follows from Lemma 3.5 that (25) equals

\[ \beta^2\gamma^2\delta\rho^2\mathbb{E}_{x,y,h_1,h_2}\Delta_{-2h_2}B(y) \Delta_{-h_2}C(x+y)D(2x+y)\Delta_{(h_2,-2h_2)}\Phi(x,y)+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg), \]

which equals

\[ \beta^4\gamma^4\delta^2\rho^4\mathbb{E}_{x,h_2}\Delta_{h_2}A(x)+O \bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg)= \alpha^2\beta^4\gamma^4\delta^2 \rho^4+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

It remains to relate

\begin{align*} \mathbb{E}_{x,y,h_1,h_2}&\Delta_{-h_1,-2h_2}B(y)\Delta_{-h_1,-h_2}C(x+y) \Delta_{-h_1}D(2x+y)\Delta_{(h_2,-2h_2)}\Phi(x,y)\\ & \cdot|\mathbb{E}_{z}\Delta_{-h_2}B(y-z)C(x+y-z)\Delta_{h_2}D(2x+y-z)\Delta_{(-h_1,h_1)}\Phi(x+z,y-2z)\\ &\Delta_{(0,h_1),(h_2,0)}f_0(x,y-2z)|^2 \end{align*}

to $\|f_0\|_{\star _1}$. Expanding the square and making a change of variables yields

\begin{align*} \mathbb{E}_{x,y,z,h_1,h_2,h_3}(&\Delta_{-h_1,-2h_2}B(y+2z)\Delta_{-h_2,-h_3}B(y+z) \\ &\Delta_{-h_1,-h_2}C(x+y+2z)\Delta_{-h_3}C(x+y+z) \\ &\Delta_{-h_1}D(2x+y+2z)\Delta_{h_2,-h_3}D(2x+y+z)\\ &\Delta_{(h_2,-2h_2)}\Phi(x,y+2z)\Delta_{(-h_1,h_1),(h_3,-2h_3)}\Phi(x+z,y) \\ &\Delta_{(0,h_1),(h_2,0),(0,-2h_3)}f_0(x,y)), \end{align*}

which can be written as

\[ \mathbb{E}_{x,y,h_1,h_2,h_3}\Delta_{(0,h_1),(h_2,0),(0,-2h_3)}f_0(x,y)\mu(x,y,h_1,h_2,h_3), \]

where

\begin{align*} \mu(x,y,h_1,h_2,h_3) := \mathbb{E}_{z}(&\Delta_{-h_1,-2h_2}B(y+2z)\Delta_{-h_2,-h_3}B(y+z) \\ &\Delta_{-h_1,-h_2}C(x+y+2z)\Delta_{-h_3}C(x+y+z) \\ &\Delta_{-h_1}D(2x+y+2z)\Delta_{h_2,-h_3}D(2x+y+z)\\ &\Delta_{(h_2,-2h_2)}\Phi(x,y+2z)\Delta_{(-h_1,h_1),(h_3,-2h_3)}\Phi(x+z,y)). \end{align*}

We will show that, for almost every $5$-tuple $(x,y,h_1,h_2,h_3)\in (\mathbb {F}_p^n)^5$ for which $x,x+h_2\in A$, $\mu (x,y,h_1,h_2,h_3)$ is very close to the constant value $\alpha ^4\beta ^8\gamma ^6\delta ^6\rho ^{6}$. Indeed, the first moment $\mathbb {E}_{\substack {x,y,h_1,h_2,h_3\\ x,x+h_2\in A}}\mu (x,y,h_1,h_2,h_3)$ is

\begin{align*} \frac{1}{\alpha^2+O(\varepsilon)}\mathbb{E}_{x,y,z,h_1,h_2,h_3}(&\Delta_{-h_1,-2h_2}B(y+2z)\Delta_{-h_2,-h_3}B(y+z) \\ &\Delta_{-h_1,-h_2}C(x+y+2z)\Delta_{-h_3}C(x+y+z) \\ &\Delta_{-h_1}D(2x+y+2z)\Delta_{h_2,-h_3}D(2x+y+z)\\ &\Delta_{(h_2,-2h_2)}\Phi(x,y+2z)\Delta_{(-h_1,h_1),(h_3,-2h_3)}\Phi(x+z,y)). \end{align*}

Lemmas 3.6 and 4.1 tell us that

\[ \mathbb{P}((x,z,h_1,h_2,h_3)\in(\mathbb{F}_p^n)^5:\|H(x,z,h_1,h_2,h_3,\cdot)-\beta^8 \gamma^6\delta^6\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon} \]

and

\begin{align*} \mathbb{P}(&(x,x+h_2,x+z,x+z-h_1,x+z+h_3,x+z-h_1+h_3)\in A^6\\ &:\operatorname{codim}\{y\in\mathbb{F}_p^n:\Delta_{(h_2,-2h_2)}\Phi(x,y+2z) \Delta_{(-h_1,h_1),(h_3,-2h_3)}\Phi(x+z,y)=1\}\neq 6d)\ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \end{align*}

where

\begin{align*} H(x,y,h_1,h_2,h_3,z) := (&\Delta_{-h_1,-2h_2}B(y+2z)\Delta_{-h_2,-h_3}B(y+z)\Delta_{-h_1,-h_2}C(x+y+2z) \\ &\Delta_{-h_3}C(x+y+z)\Delta_{-h_1}D(2x+y+2z)\Delta_{h_2,-h_3}D(2x+y+z)), \end{align*}

so it follows from Lemma 3.5 that the first moment equals

\[ \frac{1}{\alpha^2+O(\varepsilon)}\beta^8\gamma^6\delta^6 \rho^6\mathbb{E}_{x,z,h_1,h_2,h_3}\Delta_{h_2}A(x) \Delta_{-h_1,h_3}A(x+z)+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg), \]

which equals

\[ \alpha^4\beta^8\gamma^6\delta^6\rho^6+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg) \]

by Corollary 3.3. The second moment $\mathbb {E}_{\substack {x,y,h_1,h_2,h_3 \\ x,x+h_2\in A}}\mu (x,y,h_1,h_2,h_3)^2$ is

\begin{align*} \frac{1}{\alpha^2+O(\varepsilon)}\mathbb{E}_{x,y,z,h_1,h_2,h_3,k} (&\Delta_{-h_1,-2h_2,2k}B(y+2z)\Delta_{-h_2,-h_3,k}B(y+z)\\ &\Delta_{-h_1,-h_2,2k}C(x+y+2z)\Delta_{-h_3,k}C(x+y+z)\\ &\Delta_{-h_1,2k}D(2x+y+2z)\Delta_{h_2,-h_3,k}D(2x+y+z) \\ &\Delta_{(h_2,-2h_2),(0,2k)}\Phi(x,y+2z)\Delta_{(-h_1,h_1),(h_3,-2h_3),(k,0)}\Phi(x+z,y)), \end{align*}

which, noting that

\[ \Delta_{(h_2,-2h_2),(0,2k)}\Phi(x,y+2z)=\Delta_{(h_2,-2h_2)} \Phi(x,y+2z)\Delta_{(h_2,-2h_2)}\Phi(x,2k-u), \]

we can write as

\begin{align*} \mathbb{E}_{x,y,z,h_1,h_2,h_3,k}(&\Delta_{-h_1,-2h_2,2k}B(y+2z)\Delta_{-h_2,-h_3,k}B(y+z)\\ &\Delta_{-h_1,-h_2,2k}C(x+y+2z)\Delta_{-h_3,k}C(x+y+z)\\ &\Delta_{-h_1,2k}D(2x+y+2z)\Delta_{h_2,-h_3,k}D(2x+y+z) \\ &\Delta_{(h_2,-2h_2)}\Phi(x,y+2z)\Delta_{(-h_1,h_1),(h_3,-2h_3),(k,0)}\Phi(x+z,y) \\ &\Delta_{(h_2,-2h_2)}\Phi(x,2k-u)), \end{align*}

Lemmas 3.6 and 4.1, analogously to the case of the first moment, tell us that

\[ \mathbb{P}((x,z,h_1,h_2,h_3,k)\in(\mathbb{F}_p^n)^6:\|I(x,z,h_1,h_2,h_3,k,\cdot)-\beta^{16} \gamma^{12}\delta^{12}\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon} \]

and

\begin{align*} \mathbb{P}\bigg(&x\in A\cap (A-h_2)\text{ and }x+z\in \bigcap_{\omega\in\{0,1\}^8}(A-\omega\cdot(-h_1,h_3,k))\\ &:\operatorname{codim}\{y\in\mathbb{F}_p^n:\Delta_{(h_2,-2h_2)}\Phi(x,y+2z)\Delta_{(-h_1,h_1), (h_3,-2h_3),(k,0)}\Phi(x+z,y)=1\}\neq 10d\bigg)\\ &\quad \ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \end{align*}

where

\begin{align*} I(x,z,h_1,h_2,h_3,k,y) := (&\Delta_{-h_1,-2h_2,2k}B(y+2z)\Delta_{-h_2,-h_3,k}B(y+z)\\ &\Delta_{-h_1,-h_2,2k}C(x+y+2z)\Delta_{-h_3,k}C(x+y+z)\\ &\Delta_{-h_1,2k}D(2x+y+2z)\Delta_{h_2,-h_3,k}D(2x+y+z)), \end{align*}

so it follows from Lemma 3.5 that the second moment equals $(\alpha ^2+O(\varepsilon ))^{-1}$ times

(26)\begin{equation} \beta^{16}\gamma^{12}\delta^{12}\rho^{10}\mathbb{E}_{x,z,h_1,h_2,h_3,k} \Delta_{-h_1,h_3,k}A(x+z)\Delta_{(h_2,-2h_2)}\Phi(x,2k-u)+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \end{equation}

To estimate the main term of (26), we note that

\[ \|\Phi(\cdot,2\cdot-u)-\alpha\rho\|_{U^6(\mathbb{F}_p^n\times\mathbb{F}_p^n)}= \|\Phi-\alpha\rho\|_{U^6(\mathbb{F}_p^n\times\mathbb{F}_p^n)} \]

and apply Lemmas 3.6 and 4.1 again to get that

\[ \mathbb{P}((x,z,h_1,h_3)\in(\mathbb{F}_p^n)^4:\|J(x,z,h_1,h_3,\cdot)-\alpha^4\|_{U^2(\mathbb{F}_p^n)}>\varepsilon^{1/8})\ll\sqrt{\varepsilon} \]

and

\[ \mathbb{P}((x,x+h_2)\in A\times A:\operatorname{codim}\{k\in\mathbb{F}_p^n:\Delta_{(h_2,-2h_2)}\Phi(x,2k-u)=1\}\neq 2d) \ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \]

where

\[ J(x,z,h_1,h_3,k):=\Delta_{-h_1,h_3}A(x+z+k). \]

Thus, by Lemma 3.5, we have that

\[ \mathbb{E}_{x,z,h_1,h_2,h_3,k}\Delta_{-h_1,h_3,k}A(x+z)\Delta_{(h_2,-2h_2)}\Phi(x,2k-u) \]

equals

\[ \alpha^4\rho^2\mathbb{E}_{x,z,h_1,h_2,h_3}\Delta_{h_2}A(x) \Delta_{-h_1,h_3}A(x+z)+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg), \]

which equals

\[ \alpha^{10}\rho^2+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg) \]

by Corollary 3.3. It therefore follows that the second moment is

\[ \alpha^{8}\beta^{16}\gamma^{12}\delta^{12}\rho^{12}+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

Thus,

\[ \mathbb{E}_{\substack{x,y,h_1,h_2,h_3 \\ x,x+h_2\in A}} |\mu(x,y,h_1,h_2,h_3)-\alpha^4\beta^8\gamma^6\delta^6\rho^6|^2\ll\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}, \]

and we conclude that

\[ \mathbb{E}_{x,y,h_1,h_2,h_3}\Delta_{(0,h_1),(h_2,0),(0,-2h_3)}f_0(x,y) \mu(x,y,h_1,h_2,h_3) = \alpha^4\beta^8\gamma^6\delta^6\rho^6\|f_0\|_{\star_1}^8+O \bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

Putting everything together gives

\[ |\Lambda(f_0,f_1,f_2,f_3)|^8\leq\alpha^{14}\beta^{20}\gamma^{16} \delta^{16}\rho^{18}\|f_0\|_{\star_1}^8+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg), \]

as desired.

For (17), we make a change of variables and apply the Cauchy–Schwarz inequality to bound $|\Lambda (T,f_1,f_2,f_3)|^2$ by

\begin{align*} \mu_{\mathbb{F}_p^n\times\mathbb{F}_p^n}(T)\cdot\mathbb{E}_{x,y} & B(y)C(x+y) \Phi(x,y)\\ & \cdot |\mathbb{E}_zC(x+y-2z)D(2x+y-2z)f_1(x,y-z)f_3(x+z,y-2z)|^2. \end{align*}

Expanding the square in the second quantity and making a change of variables yields

\begin{align*} \mathbb{E}_{x,y,z,h_1}&B(y+2z)C(x+y+z)\Delta_{-2h_1}C(x+y-z)\Delta_{-2h_1}D(2x+y-2z)\Phi(x-z,y+2z) \\ &\Delta_{(0,-h_1)}f_1(x-z,y+z)\Delta_{(h_1,-2h_1)}f_3(x,y). \end{align*}

By applying the Cauchy–Schwarz inequality again in the $x,y,$ and $h_1$ variables, the modulus squared of this is bounded above by

(27)\begin{equation} \mathbb{E}_{x,y,h_1}\Delta_{-2h_1}B(y)\Delta_{-h_1}C(x+y)D(2x+y)\Delta_{(h_1,-2h_1)}\Phi(x,y) \end{equation}

times

\begin{align*} \mathbb{E}_{x,y,h_1}(&\Delta_{-2h_1}B(y)D(2x+y)\Delta_{(h_1,-2h_1)}\Phi(x,y)\\ &|\mathbb{E}_{z}[B(y+2z)C(x+y+z)\Delta_{-2h_1}C(x+y-z)\Delta_{-2h_1}D(2x+y-2z)\Phi(x-z,y+2z)\\ &\ \ \ \ \ \Delta_{(0,-h_1)}f_1(x-z,y+z)]|^2). \end{align*}

Expanding the square and making a change of variables, the second factor equals

\begin{align*} \mathbb{E}_{x,y,z,h_1,h_2}&\Delta_{2h_2}B(y+z)\Delta_{-2h_1}B(y-z) \Delta_{-2h_1,-h_2}C(x+y-z)\Delta_{h_2}C(x+y+z)\\ &\Delta_{-2h_1,-2h_2}D(2x+y-z)D(2x+y+z)\Delta_{(-h_2,h_2)} \Phi(x,y+z)\Delta_{(h_1,-2h_1)}\Phi(x+z,y-z) \\ &\Delta_{(0,-h_1),(-h_2,h_2)}f_1(x,y), \end{align*}

which we can write as

\[ \mathbb{E}_{x,y,h_1,h_2}\Delta_{(0,-h_1),(-h_2,h_2)}f_1(x,y)\mu'(x,y,h_1,h_2), \]

where

\begin{align*} \mu'(x,y,h_1,h_2) := \mathbb{E}_{z}&\Delta_{2h_2}B(y+z)\Delta_{-2h_1}B(y-z) \Delta_{-2h_1,-h_2}C(x+y-z)\Delta_{h_2}C(x+y+z)\\ &\Delta_{-2h_1,-2h_2}D(2x+y-z)D(2x+y+z)\Delta_{(-h_2,h_2)}\Phi(x,y+z)\\ &\Delta_{(h_1,-2h_1)}\Phi(x+z,y-z). \end{align*}

The quantity (27) and the weight $\mu '$ can be analyzed in the same manner as the corresponding quantity and weight in the proof of (16), so that

\[ |\Lambda(T,f_1,f_2,f_3)|^4\leq \alpha^{8}\beta^{8}\gamma^{10}\delta^{8} \rho^{8}\|f_1\|_{\star_2}^4+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg). \]

Finally, for (18), we make a change of variables and apply the Cauchy–Schwarz inequality to bound $|\Lambda (T,T,f_2,f_3)|^2$ by $\mu _{\mathbb {F}_p^n\times \mathbb {F}_p^n}(T)$ times

\begin{align*} \mathbb{E}_{x,y}B(y)C(x+y)\Phi(x,y)|\mathbb{E}_z &B(y+z)C(x+y-z)D(2x+y-2z)D(2x+y-z)\\ &\Phi(x-z,y+z)f_2(x-z,y+2z)|^2. \end{align*}

Expanding the square in the second quantity and making a change of variables yields

\begin{align*} \mathbb{E}_{x,y,z,h_1}&B(y)C(x+y-z)\Phi(x+z,y-2z)\Delta_{h_1}B(y-z)\Delta_{-h_1}C(x+y-2z)\\ &\Delta_{-2h_1}D(2x+y-2z)\Delta_{-h_1}D(2x+y-z)\Delta_{(-h_1,h_1)} \Phi(x,y-z)\Delta_{(-h_1,2h_1)}f_2(x,y), \end{align*}

which can be written as

\[ \mathbb{E}_{x,y,h_1}\Delta_{(-h_1,2h_1)}f_2(x,y)\mu''(x,y,h_1), \]

where

\begin{align*} \mu''(x,y,h_1):=\mathbb{E}_{z}&B(y)C(x+y-z)\Phi(x+z,y-2z)\Delta_{h_1}B(y-z)\Delta_{-h_1}C(x+y-2z)\\ &\Delta_{-2h_1}D(2x+y-2z)\Delta_{-h_1}D(2x+y-z)\Delta_{(-h_1,h_1)}\Phi(x,y-z). \end{align*}

This weight can again be analyzed in the same manner as the weights appearing in the proof of (16), giving

\[ |\Lambda(T,T,f_2,f_3)|^2\leq \alpha\beta^3\gamma^3\delta^4 \rho^3\|f_2\|_{\star_3}^2+O\bigg(\frac{\varepsilon^{\Omega(1)}}{\rho^{O(1)}}\bigg), \]

and completing the proof of the lemma.