Proof of the conclusion from (i)

Suppose that the collection ${\mathcal P}$ is of type I. Then $i=\deg (P_1)<\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_i({\mathcal P}),\ldots , v_j({\mathcal P}), 0, 0, \ldots )$ . To establish Equation (6.23), we consider three cases. Let $P\in {\mathcal P}$ . If $\deg (P)>i$ , then

(6.26) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=\deg(P), \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P), \end{aligned} \end{align} $$

which yields that $v_k({\mathcal P}')=v_k({\mathcal P})$ for all $k>i$ . If $\deg (P)=i$ and $\ell (P)\neq \ell (P_1)$ , then

(6.27) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=i, \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P)- \ell(P_1). \end{aligned} \end{align} $$

If $\deg (P)=i$ and $\ell (P)=\ell (P_1)$ , then

$$ \begin{align*} \deg(P-P_1)<i, \quad\text{ and } \quad \deg(P(\cdot+h)-P_1)<i. \end{align*} $$

The latter two cases show that $v_{k}({\mathcal P}')\ge 0$ for all and $v_i({\mathcal P}')=v_i({\mathcal P})-1$ . Hence, Equation (6.23) holds. We now show that ${\mathcal P}'$ is $(d, j)$ -admissible.

We begin with verifying Equation (6.6) for $P'\in {\mathcal P}_j^{\prime }$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ and $\varepsilon \in \{0, 1\}$ . By Equations (6.26) and (6.6) for $P\in {\mathcal P}_j$ , we obtain

(6.28) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(P')| \leq A_0\delta^{-A}N^{d-j}. \end{align} $$

We now verify Equation (6.7) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ with $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ . We may write

(6.29) $$ \begin{align} Q_1^{\prime}=Q_1(\cdot+\varepsilon_1 h)-P_1, \qquad \text{ and } \qquad Q_2^{\prime}=Q_2(\cdot+\varepsilon_2 h)-P_1 \end{align} $$

for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ . By Equation (6.26), we have $\ell (Q_1^{\prime })=\ell (Q_1)$ and $\ell (Q_2^{\prime })=\ell (Q_2)$ . Then $\ell (Q_1)\neq \ell (Q_2)$ and by Equation (6.7) for $Q_1, Q_2\in {\mathcal P}_j$ , we deduce

(6.30) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell(Q_1^{\prime}) - \ell(Q_2^{\prime})| \leq A_0\delta^{-A} N^{d-j}. \end{align} $$

We finally verify Equation (6.8) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ as in Equation (6.29) such that $Q_1^{\prime }\neq Q_2^{\prime }$ and $\ell (Q_1^{\prime })=\ell (Q_2^{\prime })=\ell $ . By Equation (6.26), we see that $\ell (Q_1)=\ell (Q_2)=\ell $ . Since ${\mathcal P}$ is $(d, j)$ -admissible, using Equation (6.6), we also have

(6.31) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j} \leq |\ell| \leq A_0\delta^{-A}N^{d-j}. \end{align} $$

Recall that $\delta '\delta ^2N/C^2\le |h| \le \delta 'N$ , where $\delta '>0$ is an arbitrarily small number such that $\delta '\le \delta ^4/C$ . Set $\delta ':=\delta ^{M}(CM)^{-1}$ for a large number $M\ge 1$ , which will be chosen later.

First, suppose $Q_1 = Q_2$ . Then $\varepsilon _1 \not = \varepsilon _2$ and $\deg (Q_1^{\prime }-Q_2^{\prime })=j-1$ . Furthermore, $\ell (Q_1^{\prime }-Q_2^{\prime })= j\ell h(\varepsilon _1-\varepsilon _2)$ implying $|\ell (Q_1^{\prime } - Q_2^{\prime })| = |j \ell h|$ , and so by Equation (6.31),

(6.32) $$ \begin{align} |j|(A_0C^3M)^{-1}\delta^{A+M+2} N^{d-j+1}\le |j\ell h| \le |j| A_0(CM)^{-1}\delta^{M-A} N^{d-j+1}, \end{align} $$

and this verifies Equations (6.8) in the case $Q_1 = Q_2$ .

Now, suppose $Q_1 \not = Q_2$ so that $\deg (Q_1 - Q_2) = j-1$ and Equation (6.8) holds for $\ell (Q_1 - Q_2)$ ; that is,

(6.33) $$ \begin{align} A_0^{-1}\delta^{A} N^{d-j+1} \leq |\ell(Q_1-Q_2)| \leq A_0\delta^{-A} N^{d-j+1}. \end{align} $$

Taking $M:=\max \{2A, 2|j|A_0^2\}$ in Equation (6.32), we see that $|j\ell h| \le \frac 12 A_0^{-1}\delta ^{A} N^{d-j+1}$ if $C> 1$ is large enough.

In this case, $\ell (Q_1^{\prime } - Q_2^{\prime }) = \ell (Q_1 - Q_2) + j h\ell (\varepsilon _1-\varepsilon _2)$ and so

$$ \begin{align*} |\ell(Q_1-Q_2)|-|j\ell h|\le |\ell(Q_1^{\prime}-Q_2^{\prime})| \le |\ell(Q_1-Q_2)|+|j\ell h|. \end{align*} $$

From Equation (6.33) and $|j\ell h| \le \frac 12 A_0^{-1} \delta ^A N^{d-j+1}$ , we conclude

(6.34) $$ \begin{align} \frac{1}{2}A_0^{-1}\delta^{A} N^{d-j+1} \leq |\ell(Q_1^{\prime}-Q_2^{\prime})| \leq \frac{3}{2}A_0\delta^{-A} N^{d-j+1}. \end{align} $$

This verifies Equation (6.8) in the case $Q_1 \not = Q_2$ .

In either case, we see that $\deg (Q_1^{\prime } - Q_2^{\prime }) = j-1$ and (see Equations (6.32) and (6.34)) we can find a tolerance pair $(A_0^{\prime }, A')$ for ${\mathcal P}'$ depending on the tolerance $(A_0, A)$ of ${\mathcal P}$ and the constants C and M such that

(6.35) $$ \begin{align} (A_0^{\prime})^{-1}\delta^{A'} N^{d-j+1} \leq |\ell(Q_1^{\prime}-Q_2^{\prime})| \leq A_0^{\prime}\delta^{-A'} N^{d-j+1} \end{align} $$

holds, establishing Equation (6.8).

Proof of the conclusion from (ii)

Suppose that the collection ${\mathcal P}$ is of type II. Then $\deg (P_1)=\ldots =\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_j({\mathcal P}), 0, 0, \ldots )$ with $v_j({\mathcal P})>1$ . To establish Equation (6.24), we will proceed in a similar way as in (i). If $P\in {\mathcal P}={\mathcal P}_j$ and $\ell (P)\neq \ell (P_1)$ , then

(6.36) $$ \begin{align} \begin{aligned} \deg(P-P_1)=\deg(P(\cdot+h)-P_1)=j, \\ \ell(P-P_1)=\ell(P(\cdot+h)-P_1)=\ell(P)- \ell(P_1). \end{aligned} \end{align} $$

If $P\in {\mathcal P}={\mathcal P}_j$ and $\ell (P)=\ell (P_1)$ , then by the fact that ${\mathcal P}$ is $(d, j)$ -admissible, and by Equation (6.8), we see that

(6.37) $$ \begin{align} \deg(P-P_1)<j, \quad\text{ and } \quad \deg(P(\cdot+h)-P_1)<j. \end{align} $$

This shows that $v_{k}({\mathcal P}')\ge 0$ for all and $v_j({\mathcal P}')=v_j({\mathcal P})-1$ . Hence, Equation (6.24) holds. We now show that ${\mathcal P}'$ is $(d, j)$ -admissible.

We begin with verifying Equation (6.6) for $P'\in {\mathcal P}_j^{\prime }$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ such that $\ell (P)\neq \ell (P_1)$ and $\varepsilon \in \{0, 1\}$ . Since ${\mathcal P}$ is $(d, j)$ -admissible, using Equations (6.36) and (6.7) (with $\ell (P)- \ell (P_1)$ in place of $\ell (P)- \ell (Q)$ ), we obtain Equation (6.28) which is Equation (6.6) for $P' \in \mathcal P_j^{\prime }$ .

We now verify Equation (6.7) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ with $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ . As in Equation (6.29), we may write $Q_1^{\prime }=Q_1(\cdot +\varepsilon _1 h)-P_1$ , and $Q_2^{\prime }=Q_2(\cdot +\varepsilon _2 h)-P_1$ for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ such that $\ell (Q_1)\neq \ell (P_1)$ and $\ell (Q_2)\neq \ell (P_1)$ . By Equation (6.36), we have $\ell (Q_1^{\prime })=\ell (Q_1)-\ell (P_1)$ and $\ell (Q_2^{\prime })=\ell (Q_2)-\ell (P_1)$ . Then $\ell (Q_1)\neq \ell (Q_2)$ and Equation (6.30) is verified by appealing to Equation (6.7) (with $\ell (Q_1) - \ell (Q_2)$ in place of $\ell (P) - \ell (Q)$ ).

We finally verify Equation (6.8) for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_j$ as in Equation (6.29) such that $Q_1^{\prime }\neq Q_2^{\prime }$ and $\ell (Q_1^{\prime })=\ell (Q_2^{\prime })=\ell $ . By Equation (6.36), $\ell (Q_1)-\ell (P_1)=\ell (Q_2)-\ell (P_1)=\ell $ and since ${\mathcal P}$ is $(d,j)$ -admissible, we see that $\ell $ satisfies Equation (6.31). Now, by following the last part of the proof from (i), we conclude that Equation (6.35) holds.

Proof of the conclusion from (iii)

Suppose that the collection ${\mathcal P}$ is of type III. Then $\deg (P_1)=\ldots =\deg (P_m)=j$ and $v({\mathcal P})=(0,\ldots , 0, v_j({\mathcal P}), 0, 0, \ldots )$ with $v_j({\mathcal P})=1$ , thus $\ell (P_1)=\ldots =\ell (P_m):=\ell $ . To establish Equation (6.25), we will proceed in a similar way as in (i) and (ii). If $P\in {\mathcal P}_j$ and $\ell (P)=\ell $ , then Equation (6.31) holds for $\ell $ and once again Equation (6.37) holds. This in turn implies that $v_{j-1}({\mathcal P}')>0$ and $v_{k}({\mathcal P}')=0$ for all $k\neq j-1$ . Hence, Equation (6.25) holds. We now show that ${\mathcal P}'$ is $(d, j-1)$ -admissible.

We begin with verifying Equation (6.6) (or equivalently Equation (6.28) with j replaced by $j-1$ ) for $P'\in {\mathcal P}_{j-1}'$ . We may write $P'=P(\cdot +\varepsilon h)-P_1$ for some $P\in {\mathcal P}_j$ such that $\ell (P)=\ell (P_1)$ and $\varepsilon \in \{0, 1\}$ . Then

(6.38) $$ \begin{align} \ell(P')=\ell(P(\cdot+\varepsilon h)-P_1)= \ell(P-P_1) + j h \ell \varepsilon. \end{align} $$

As in (i) we have $\delta '\delta ^2N/C^2\le |h| \le \delta 'N$ , where $\delta ':=\delta ^{M}(CM)^{-1}$ for a large number $M\ge 1$ , which will be chosen later. Furthermore if $P \not = P_1$ , then $A_0^{-1} \delta ^A N^{d-j+1} \le |\ell (P-P_1)| \le A_0 \delta ^{-A} N^{d-j+1}$ since ${\mathcal P}$ is $(d,j)$ -admissible and so Equation (6.8) holds with $Q = P_1$ . This takes care of the case $\varepsilon = 0$ .

If $\varepsilon =1$ and $P = P_1$ , then Equation (6.32) gives the desired bound for $|\ell (P')|$ . When $P \not = P_1$ , we use the upper bound from Equation (6.32)

(6.39) $$ \begin{align} |jh\ell| \leq |j|A_0(CM)^{-1}\delta^{M-A}N^{d-j+1} \le \frac12 A_0^{-1} \delta^{-A} N^{d-j+1} \end{align} $$

when $M = \max (2A, 2|j|A_0^2)$ and $C>1$ chosen large enough. Thus, as before, condition (6.6) holds for $P'$ with some tolerance pair $(A_0^{\prime }, A')$ as desired.

For $Q_1^{\prime } \not = Q_2^{\prime }\in {\mathcal P}^{\prime }_{j-1}$ , we may write $Q_1^{\prime }=Q_1(\cdot +\varepsilon _1 h)-P_1$ , and $Q_2^{\prime }=Q_2(\cdot +\varepsilon _2 h)-P_1$ for some $Q_1, Q_2\in {\mathcal P}_j$ and $\varepsilon _1, \varepsilon _2\in \{0, 1\}$ such that $\ell (Q_1)=\ell (Q_2) = \ell (P_1) = \ell $ . We have $\ell (Q_1 - P_1) - \ell (Q_2 - P_1) = \ell (Q_1 - Q_2)$ and so by Equation (6.38),

(6.40) $$ \begin{align} \ell(Q_1^{\prime})-\ell(Q_2^{\prime})= \ell(Q_1 - Q_2) + j h \ell (\varepsilon_1 - \varepsilon_2). \end{align} $$

We consider two cases.

If $Q_1 = Q_2$ , then necessarily $|\varepsilon _1 - \varepsilon _2| = 1$ and so $\ell (Q_1^{\prime }) \not = \ell (Q_2^{\prime })$ , $\deg (Q_1^{\prime } - Q_2^{\prime }) = j-1$ , and Equation (6.32) shows that Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime } \in {\mathcal P}^{\prime }_{j-1}$ .

If $Q_1 \not = Q_2$ , then $A_0^{-1} \delta ^A N^{d-j+1} \le |\ell (Q_1-Q_2)| \le A_0 \delta ^{-A} N^{d-j+1}$ since ${\mathcal P}$ is $(d,j)$ -admissible, and so Equation (6.8) holds with $P=Q_1 $ and $Q=Q_2$ . From Equation (6.39), we see that $\ell (Q_1^{\prime }) \not = \ell (Q_2^{\prime })$ and Equation (6.40) implies that Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime }\in {\mathcal P}^{\prime }_{j-1}$ .

In either case, we see that Equation (6.8) is vacuously satisfied by ${\mathcal P}'$ and Equation (6.7) holds for $Q_1^{\prime }, Q_2^{\prime } \in {\mathcal P}^{\prime }_{j-1}$ with (necessarily) $\ell (Q_1^{\prime })\neq \ell (Q_2^{\prime })$ .

Concluding, we are able to find a tolerance pair $(A_0^{\prime }, A')$ for ${\mathcal P}'$ depending on the tolerance $(A_0, A)$ of ${\mathcal P}$ and the constants C and M such that the required estimates for Equations (6.38) and (6.40) hold. This completes the proof of Proposition 6.20.

Step 1.

Assume that $m=1$ so that $N_0 = N^{\deg {P_1}}$ . For $\zeta \in \mathbb {K}$ and $\xi =(\xi _1,\ldots , \xi _n)\in \mathbb {K}^n$ , we define the multiplier

$$ \begin{align*} m_N(\zeta, \xi):=\int_{\mathbb{K}} e\Big(-\zeta P_1(y) + \sum_{j=1}^n \xi_j Q_j(y)\Big)d\mu_{[N]}(y). \end{align*} $$

We now express

$$ \begin{align*} \Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(g_0,g_1) =N_0^{-1}\int_{\mathbb{K}} \widehat{g_0}(-\zeta) \widehat{g_1}(\zeta) m_N(\zeta, \xi)d\mu(\zeta). \end{align*} $$

Using the Cauchy–Schwarz inequality and Plancherel’s theorem, we see

(6.60) $$ \begin{align} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(g_0,g_1)|\leq N_0^{-1}\| g_0 \|_{L^2(\mathbb{K})} \|g_1\|_{L^2(\mathbb{K})} \sup_{\zeta \in \operatorname{\mathrm{supp}}{(\widehat{g_0}\widehat{g_1})}} |m_N(\zeta, \xi)|. \end{align} $$

When $\mathbb {K}$ is non-Archimedean, let so that . When $\mathbb {K}$ is Archimedean, choose a Schwartz function $\varphi : \mathbb {K} \to \mathbb {K}$ such that

For a scale M, we set $\varphi _M(x) = M^{-1} \varphi (M^{-1} x)$ when $\mathbb {K} = {\mathbb R}$ and when $\mathbb {K} = {\mathbb C}$ , we set $\varphi _{M}(z) = M^{-1} \varphi (M^{-1/2} z)$ . When $\mathbb {K}$ is non-Archimedean, we set .

Consider two scales $M_1\simeq \delta ^{C}N^{\deg {P_1}}$ and $N_1\simeq \delta ^{2C}N^{\deg {P_1}}/C$ . Then we obtain

$$ \begin{align*} \delta\le |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1)|\le |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,\varphi_{M_1}*f_1)|+ |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|. \end{align*} $$

Note that

$$ \begin{align*} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,\varphi_{M_1}*f_1)| \le N_0^{-1}\|f_0\|_{L^{\infty}(\mathbb{K})}\|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})} \le N_0^{-1} \|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})}, \end{align*} $$

and

$$ \begin{align*} \|\varphi_{M_1}*f_1\|_{L^1(\mathbb{K})}&\le \|\varphi_{M_1}*\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})}+ \|(\varphi_{M_1}-\varphi_{M_1}*\mu_{[N_1]})*f_1\|_{L^1(\mathbb{K})}\\ &\lesssim \|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})} +C^{-1}\delta^C N_0 \end{align*} $$

since $\varphi _{M_1}-\varphi _{M_1}*\mu _{[N_1]} = 0$ when $\mathbb {K}$ is non-Archimedean and when $\mathbb {K}$ is Archimedean, we have the pointwise bound

$$ \begin{align*} |\varphi_{M_1}(x)-\varphi_{M_1}*\mu_{[N_1]}(x)|\lesssim C^{-1}\delta^C \, M_1^{-1}\big(1+M_1^{-1}|x|\big)^{-10}. \end{align*} $$

If $C\ge 1$ is sufficiently large, then we may write

(6.61) $$ \begin{align} \delta\lesssim |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1)|\le N_0^{-1}\|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})}+ |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|. \end{align} $$

By Equation (6.60), we have that

(6.62) $$ \begin{align} |\Lambda_{\mathcal{P};N}^{\mathcal{Q}; \xi}(f_0,f_1-\varphi_{M_1}*f_1)|\lesssim \sup_{\zeta\in \mathbb{K}: |\zeta|\ge M_1^{-1} } |m_N(\zeta, \xi)| \end{align} $$

since $\| f_0 \|_{L^2(\mathbb {K})}\le N_0^{1/2}$ and $\|f_1\|_{L^2(\mathbb {K})}\le N_0^{1/2}$ . We now prove that

(6.63) $$ \begin{align} \sup_{\zeta\in \mathbb{K}: |\zeta|\ge M_1^{-1} } |m_N(\zeta, \xi)|\lesssim \delta^{2}. \end{align} $$

Suppose that inequality (6.63) does not hold, then one has

$$ \begin{align*} |m_N(\zeta, \xi)|\gtrsim \delta^{2} \end{align*} $$

for some $\zeta \in \mathbb {K}$ so that $|\zeta |\ge M_1^{-1}$ . Then Lemma 6.49 implies $N^{\deg {P_1}} |\zeta | \lesssim \delta ^{-A}$ for some large, fixed $A\gtrsim 1$ by Equation (6.52). Since $M_1 = \delta ^C N^{\deg {P_1}}$ , we have $\delta ^{-C} \lesssim \delta ^{-A}$ which is a contradiction if $C \gg A$ . Thus, Equation (6.63) holds.

Hence, by Equations (6.63), (6.62) and (6.61), we see that

$$ \begin{align*}\delta \lesssim N_0^{-1}\|\mu_{[N_1]}*f_1\|_{L^1(\mathbb{K})} \end{align*} $$

which establishes Theorem 6.41 when $m=1$ .

Step 2.

We now assume that Theorem 6.41 is true for $m-1$ in place of m for some integer $m\ge 2$ . Using Theorem 6.57, we show that this implies Theorem 6.41 for $m\ge 2$ . Note that bound (6.43) implies inequality (6.48) from Proposition 6.47. Now, by Theorem 6.57 applied $s-2$ times we may conclude that

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], [H_2]}^2(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}, \end{align*} $$

where $H_1, H_2\simeq \delta ^{O_{\mathcal P}(1)}N^{\deg {P_m}}$ . By Lemma 5.1, we can find a $\xi _0\in \mathbb {K}$ such that

(6.64) $$ \begin{align} N_0^{-1}\big|\widehat{F_m^{\xi}}(\xi_0)\big| \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)} \end{align} $$

since $N_0 = N^{\deg {P_m}}$ . By definitions (6.45) and (6.46) and making the change of variables $x\mapsto x-P_m(y)$ , we may write

$$ \begin{align*} N_0^{-1}\widehat{F_m^{\xi}}(\xi_0)&=N_0^{-1}\iint_{\mathbb{K}^2}F_{m;y}^{\xi}(x)e(-\xi_0 x)d\mu_{[N]}(y)d\mu(x)\\&=N_0^{-1}\iint_{\mathbb{K}^2}\mathrm{M}_{\xi_0}f_0(x)\prod_{i=1}^{m-1}f_{i}(x-P_i(y))e\Big(\xi_0 P_m(y)+\sum_{j=1}^n\xi_jQ_j(y)\Big)d\mu_{[N]}(y)d\mu(x)\\&=M^{-1} \Lambda_{{\mathcal P}'; N}^{\mathcal Q', \xi'}(\mathrm{M}_{\xi_0}f_0, f_1,\ldots, f_{m-1}), \end{align*} $$

where $\mathrm {M}_{\xi _0}f_0(x):=\mathrm {e}(-\xi _0 x)f_0(x), \, {\mathcal P}':={\mathcal P}\setminus \{P_m\}, \, \mathcal Q':=\mathcal Q\cup \{P_m\}, \, \xi ':=(\xi _0, \xi _1,\ldots , \xi _n)\in \mathbb {K}^{n+1}$ and $M:=N^{\deg (P_m)-\deg (P_{m-1})}$ . The parameter $N_0^{\prime } := N_0 M^{-1}$ is what appears in the m-linear form $\Lambda _{{\mathcal P}';N}^{\mathcal Q', \xi '}$ . We note that $N_0^{\prime } = N^{\deg {P_{m-1}}}$ .

Thus, Equation (6.64) implies

$$ \begin{align*} M^{-1} |\Lambda_{{\mathcal P}'; N, I}^{\mathcal Q', \xi'}(\mathrm{M}_{\xi_0}f_0, f_1,\ldots, f_{m-1})|\gtrsim \delta^{O(1)}. \end{align*} $$

By translation invariance, we may assume that all functions $f_0, f_1,\ldots , f_{m-1}$ are supported in $[N_0]$ . We can partition into $L \simeq M$ sets, each with measure $\simeq N_0^{\prime }$ contained in an interval $I_k$ lying in an $O(N_0^{\prime })$ neighbourhood of $E_k$ . Furthermore, $E_k$ is an $O(N_1)$ neighbourhood of a set $F_k$ such that $\mu (E_k\setminus F_k) \lesssim N_1$ and . Here, $N_1 \simeq \delta ^{O_{\mathcal P}(1)} N^{\deg (P_1)}$ . In the non-Archimedean setting, this decomposition is straightforward; in this case, we can take $F_k = E_k = I_k$ . If fact if $N_0 = q^{n_0}$ and $N_0^{\prime } = q^{n_0 - \ell }$ so that $M = q^{\ell }$ , then

$$ \begin{align*} [N_0] \ = \ B_{q^{n_0}}(0) \ = \ \bigcup_{x \in {\mathcal F}} B_{q^{n_0 -\ell}}(x) \end{align*} $$

gives our partition of $[N_0]$ , where ${\mathcal F} = \{ x = \sum _{j=0}^{\ell -1} x_j \pi ^{-n_0 + j} : x_j \in o_{\mathbb {K}}/m_{\mathbb {K}} \}$ . Note $\# {\mathcal F} = q^{\ell } = M$ . When $\mathbb {K} = {\mathbb R}$ , one simply decomposes the interval $[N_0] = [-N_0, N_0]$ into M subintervals of equal length and then extend and shrink to obtain intervals $I_k$ and $F_k$ with the desired properties.

When $\mathbb {K} = {\mathbb C}$ , the set $[N_0]$ is a disc and the decomposition is not as straightforward but not difficult to construct by starting with a mesh of squares of side length $\sqrt {N_0^{\prime }}$ which cover $[N_0]$ . It is important that for this case (when $\mathbb {K} = {\mathbb C}$ ) that we allow the sets $E_k$ and $F_k$ to be general sets (not necessarily intervals) with the above properties. The picture should be clear.

Hence, by changing variables $x \to x+P_1(y)$ and then back again,

where and .

By the pigeonhole principle, there exists such that $\# L_0\gtrsim \delta ^{O_{{\mathcal P}'}(1)}M$ , and for every $k\in L_0$ , we have

$$ \begin{align*} |\Lambda_{{\mathcal P}'; N, I_k}^{\mathcal Q', \xi'}(f_0^k, g^k, f_2^k,\ldots, f_{m-1}^k)|\gtrsim \delta^{O_{{\mathcal P}'}(1)}. \end{align*} $$

By the inductive hypothesis, we have

for every $k\in L_0$ and for every $N_1 = \delta ^C N^{\deg {P_1}}$ with $C \ge C_1({\mathcal P}')$ . Note that

(6.65)

and hence for $C\gg 1$ large enough,

Now, we can sum over $k\in L_0$ , using the bound $\# L_0\gtrsim \delta ^{O(1)}M$ and the pairwise disjoint supports of , we obtain

which by (6.65) yields

$$ \begin{align*} N_0^{-1}\big\|\mu_{[N_1]}*f_1\big\|_{L^1(\mathbb{K})} \gtrsim \ \delta^{O_{\mathcal P}(1)}, \end{align*} $$

as desired.

We now state two auxiliary technical lemmas which will be needed in the proof of Theorem 6.57. For $\omega = (\omega _1, \ldots , \omega _n) \in \{0,1\}^n$ and $h = (h_1, \ldots , h_n) \in \mathbb {K}^n$ , we write $\omega \cdot h = \sum _{i=1}^n \omega _i h_i$ and $1-\omega = (1-\omega _1, \ldots , 1-\omega _n)$ .

Lemma 6.66. Let $N\ge 1$ be a scale, and let $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ , $n\in \mathbb {N}$ and scales $H_1,\ldots , H_n$ with each $H_i\le N$ be given. Assume that $\phi :X\to \mathbb R$ is a measurable function defined on a measurable set $X\subseteq H:=\prod _{i=1}^n [H_i]$ . Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials. For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . Suppose that

(6.67)

Then

(6.68) $$ \begin{align} \int_{\square_n(X)}\bigg|N_0^{-1}\int_{\mathbb{K}} F_m(x; h, h')e(-\psi(h, h')x)d\mu(x)\bigg|^2 d\Big(\bigotimes_{i=1}^{n}\nu_{[H_i]}\Big)^{\otimes2}(h, h') \gtrsim \delta^{O(1)}, \end{align} $$

where

$$ \begin{align*} \square_n(X):=\Big\{(h, h')\in H^2: \omega \cdot h + (1-\omega)\cdot h' \in X \text{ for every } \omega\in\{0, 1\}^n\Big\}, \end{align*} $$

and

$$ \begin{align*} F_m(x; h, h'):=\int_{\mathbb{K}}{\Delta}_{h'-h}^nf_0(x+P_m(y))\prod_{i=1}^{m-1} {\Delta}_{h'-h}^nf_{i}(x-P_i(y)+P_m(y))d\mu_{[N]}(y),\\ \psi(h, h'):=\sum_{\omega\in\{0, 1\}^n}(-1)^{|\omega|}\phi\Big(\omega \cdot h + (1-\omega)\cdot h'\Big). \end{align*} $$

Proof. We shall write ${\boldsymbol{\nu }}_n:=\bigotimes _{i=1}^{n}\nu _{[H_i]}$ . Using Equations (4.2), (6.45) and (6.46), we see that the left-hand side of Equation (6.67) can be written as

$$ \begin{align*} \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n+1}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^n}G_0(x, z, h;\boldsymbol{y}) d{\boldsymbol{\nu }}_n(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n+1}}_{[N]}(\boldsymbol{y}), \end{align*} $$

where for $\boldsymbol{y}=(y_{(\omega , 0)}, y_{(\omega , 1)})_{\omega \in \{0, 1\}^n}\in \mathbb {K}^{2^{n+1}}$ , $x, z\in \mathbb {K}$ and $h\in \mathbb {K}^n$ . We have set

Write elements in X as $(h_1, h)$ with $h_1 \in \mathbb {K}$ , and apply the Cauchy–Schwarz inequality in all but the $h_1$ variable (noting that $(x,z) \to G_0(x, z, h;\boldsymbol{y})$ is supported in a product of intervals of measure $\simeq N_0^2$ ) to conclude

(6.69) $$ \begin{align} \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^{n-1}} H_0(x, z, h;\boldsymbol{y}) d{\boldsymbol{\nu }}_{n-1}(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n}}_{[N]}(\boldsymbol{y})\gtrsim \delta^{O(1)}, \end{align} $$

where

$$ \begin{align*} H_0(x, z, h;\boldsymbol{y}):=\bigg|\int_{\mathbb{K}}G_0^1(x, z, (h_1, h);\boldsymbol{y})d\nu_{[H_1]}(h_1)\bigg|^2, \end{align*} $$

and

for $\boldsymbol{y}=(y_{(1, \omega , 0)}, y_{(1, \omega , 1)})_{(j,\omega )\in \{0, 1\}^{n}}\in \mathbb {K}^{2^{n}}$ and $x, z, h_1\in \mathbb {K}$ , $h\in \mathbb {K}^{n-1}$ . Expanding the square and changing variables $x\mapsto x-h_1$ and $z\mapsto z-h_1$ , we may rewrite Equation (6.69) as

(6.70) $$ \begin{align} \nonumber \frac{1}{N_0^2} \int_{\mathbb{K}^{2^{n}}}\int_{\mathbb{K}^2}\int_{\mathbb{K}^{n+1}} G_1(x, z, h_1, h_1^{\prime}, h;\boldsymbol{y})d\nu_{[H_1]}^{\otimes2}&(h_1, h_1^{\prime}) d{\boldsymbol{\nu }}_{n-1}(h)d\mu(x)d\mu(z)d\mu^{\otimes 2^{n}}_{[N]}(\boldsymbol{y})\\&\gtrsim \delta^{O(1)}, \end{align} $$

where

Iteratively, for each $i\in \{2,\ldots , n\}$ , we apply the Cauchy–Schwarz inequality in all but the $h_i$ variable to conclude that

$$ \begin{align*} \frac{1}{N_0^2}\int_{\mathbb{K}^2}\int_{\mathbb{K}^2}\int_{\square_n(X)} G_n(x, z, h, h';y, y') d{\boldsymbol{\nu }}_n^{\otimes2}(h, h')d\mu(x)d\mu(z)d\mu^{\otimes 2}_{[N]}(y, y') \gtrsim \delta^{O(1)}, \end{align*} $$

where

$$ \begin{align*} G_n(x, z, h, h';y, y'):={\Delta}_{h'-h}^nF_{m; y}^{\xi}(x)\overline{{\Delta}_{h'-h}^nF_{m; y'}^{\xi}(z)} \mathrm{e}(-\psi((h, h'))(x-z)). \end{align*} $$

We have arrived at Equation (6.68), completing the proof of the lemma

The following lemma is a slight variant of a result found in [Reference Peluse38].

Lemma 6.71. Given a scale $N\ge 1$ , $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ , $n\in \mathbb {N}$ and scales $H_1,\ldots , H_{n+1}$ with each $H_i \le N$ . We assume for every that $\varphi _i:\mathbb {K}^n\to \mathbb {K}$ is a measurable function independent of the variable $h_i$ in a vector $h=(h_1, \ldots , h_n)\in \mathbb {K}^n$ . Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials. For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 = N^{\deg {P_m}}$ . Suppose that

(6.72)

Then

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1],\ldots, [H_{n+1}]}^{n+1}(I)} \gtrsim_{\mathcal P} \delta^{O_{\mathcal P}(1)}. \end{align*} $$

Proof. We shall write as before ${\boldsymbol{\nu }}_n:=\bigotimes _{i=1}^{n}\nu _{[H_i]}$ and also let ${\boldsymbol{\mu}}_n:=\bigotimes _{i=1}^n\mu _{[H_i]}$ . Expanding the Fejér kernel, we may write the left-hand side of Equation (6.72) as

We apply the Cauchy–Schwarz inequality in the $x, z$ and $h'$ variables and Corollary 4.7 to deduce that

$$ \begin{align*} {\mathcal I}^{2^n}&\le \frac{1}{N_0^{2}}\int_{\mathbb{K}^{3n+2}} \prod_{\omega\in\{0, 1\}^n}\mathcal C^{|\omega|}\big({\Delta}_{h^{(\omega)}-h}^nF_m^{\xi}(x) \overline{{\Delta}_{h^{(\omega)}-h}^nF_m^{\xi}(z)}\big)d{\boldsymbol{\mu}}_n^{\otimes3}(h^{(0)}, h^{(1)}, h)d\mu(x)d\mu(z)\\ =& \frac{1}{N_0^2} \int_{\mathbb{K}^{3n}} {\mathcal A}(x,z, h_n^{\prime}, h^{(0)}, h^{(1)}, h'){{\mathcal B}}(x,z,h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n-1}^{\otimes3}(h^{(0)}, h^{(1)}, h')d\mu_{[H_n]}(h_n^{\prime}), \end{align*} $$

where

$$ \begin{align*} {\mathcal A}(x,z,h_n^{\prime}, h^{(0)}, h^{(1)}, h') :=& \int_{\mathbb{K}^2} \prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( F_m^{\xi}(x + h_n^0 - h_n^{\prime})\\ &\times{\overline{F_m^{\xi}(x+h_n^1 - h_n^{\prime}) F_m^{\xi}(z+ h_n^0 - h_n^{\prime})}} F_m^{\xi}(z + h_n^1 - h_n^{\prime}) \Big)\Big] d\mu_{[H_n]}^{\otimes2}(h_n^0, h_n^1), \end{align*} $$

$$ \begin{align*} {\mathcal B}(x,z,h^{(0)}, h^{(1)}, h'):=\prod_{\omega'\in\{0, 1\}^{n-1}} {\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( |F_m^{\xi}(x)|^2 |F_m^{\xi}(z)|^2 \Big)\Big]. \end{align*} $$

Since ${\mathcal A}\ge 0$ , we see that

$$ \begin{align*} {\mathcal I}^{2^n} \le &N_0^{-2} \int_{\mathbb{K}^{3n}} {\mathcal A}(x,z, h_n^{\prime}, h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n-1}^{\otimes3}(h^{(0)}, h^{(1)}, h')d\mu_{[H_n]}(h_n^{\prime})\\ &= \frac{1}{N_0^2} \int_{\mathbb{K}^{3n+1}} \prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} \Big( F_m^{\xi}(x + h_n^0)\overline{F_m^{\xi}(x+h_n^1)}\\ &\hspace{3cm} \times \overline{F_m^{\xi}(z+ h_n^0)} F_m^{\xi}(z + h_n^1) \Big)\Big] d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h^{(0)}, h^{(1)}) d{\boldsymbol{\mu}}_{n-1}(h')\\ &= \frac{1}{N_0^2} \int_{\mathbb{K}^{3n+1}} \mathcal C(x,z,h^{(0)}, h^{(1)}, h') d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h^{(0)}, h^{(1)}) d{\boldsymbol{\mu}}_{n-1}(h'), \end{align*} $$

where

$$\begin{align*}\mathcal C(x,z,h^{(0)}, h^{(1)}, h'):=\prod_{\omega'\in\{0, 1\}^{n-1}}{\mathcal C}^{|\omega'|} \Big[{\Delta}_{h^{(\omega')}-h'}^{n-1} {\Delta}_{h_n^1 - h_n^0}\Big( F_m^{\xi}(x) {\overline{F_m^{\xi}(z)}}\Big)\Big]. \end{align*}$$

In the penultimate equality, we made the change of variables $x \to x -h_n^0+h_n^{\prime }$ and $z \to z - h_n^0+ h_n^{\prime }$ . Now, proceeding inductively we see that

$$ \begin{align*}{\mathcal I}^{2^n} \le \frac{1}{N_0^2}\int_{\mathbb{K}^{2n+2}} {\Delta}_{h-h'}^n F_m^{\xi}(x) {\overline{{\Delta}^n_{h-h'} F_m^{\xi}(z)}} d\mu(x) d\mu(z) d{\boldsymbol{\mu}}_{n}^{\otimes2}(h, h'). \end{align*} $$

Inserting an extra average in the x variable and using the pigeonhole principle to fix z, it follows that

$$ \begin{align*} {\mathcal I}^{2^n}\le \frac{1}{N_0}\int_{\mathbb{K}^{2n+1}} \overline{{\Delta}_{h-h'}^nF_m^{\xi}(z)}\int_{\mathbb{K}}{\Delta}_{h-h'}^nF_m^{\xi}(x+w)d\mu_{[H_{n+1}]}(w)d{\boldsymbol{\mu}}_n^{\otimes2}(h,h')d\mu(x). \end{align*} $$

To conclude, we apply the Cauchy–Schwarz inequality to double the w variable and so

$$ \begin{align*}\delta^{2^{n+1}} \le {\mathcal I}^{2^{n+1}} \le \frac{1}{N_0} \int_{\mathbb{K}^{2n+3}} {\Delta}_{h-h'}^{n+1} F_m^{\xi}(x) d{\boldsymbol{\mu}}_{n+1}^{\otimes2}(h,h')d\mu(x) = \|F_m^{\xi}\|_{\square_{[H_1],\ldots, [H_{n+1}]}^{n+1}(I)}. \end{align*} $$

This completes the proof of the lemma.

Proof of Theorem 6.57.

The proof is by induction on $m\in \mathbb Z_+$ . The proof will consist of several steps. We begin by establishing the following claim.

Claim 6.73. Let $N\ge 1$ be a scale, $0<\delta \le 1$ , $m\in \mathbb Z_+$ with $m\ge 2$ and $n\in \mathbb {N}$ be given. Let ${\mathcal P}:=\{P_1,\ldots , P_{m}\}$ and $\mathcal Q:=\{Q_1,\ldots , Q_n\}$ be collections of polynomials such that

$$ \begin{align*}1\le \deg{P_1}<\ldots<\deg{P_{m}}<\deg{Q_1}<\ldots<\deg{Q_n}. \end{align*} $$

For $\xi \in \mathbb {K}^n$ , let $F_m^{\xi }$ be the dual function defined in Equation (6.45) that corresponds to the form (6.42) and $1$ -bounded functions $f_0, f_1,\ldots , f_{m-1}\in L^0(\mathbb {K})$ supported on an interval $I\subset \mathbb {K}$ of measure $N_0 := N^{\deg {P_m}}$ . Suppose that

(6.74) $$ \begin{align} N^{-1}_0\big|\widehat{F_m^{\xi}}(\zeta)\big| \ge \delta. \end{align} $$

Then for any sufficiently large constant $C\gtrsim _{{\mathcal P}, \mathcal Q}1$ one has

(6.75)

The proof of Claim 6.73 for each integer $m\ge 2$ is itself part of the inductive proof of Theorem 6.57. In the first step, we prove Claim 6.73 for $m=2$ . In the second step, we show that Claim 6.73 for all integers $m\ge 2$ implies Theorem 6.57, this in particular will establish Theorem 6.57 for $m=2$ . In the third step, we finally show that Claim 6.73 for all integers $m\ge 3$ follows from Claim 6.73 and Theorem 6.57 for $m-1$ . Taken together, this shows that Claim 6.73 and Theorem 6.57 hold for each integer $m\ge 2$ , completing the proof of Theorem 6.57.

Step 1.

We now prove Claim 6.73 for $m=2$ . Here, $N_0 = N^{\deg {P_2}}$ . For $\zeta _1, \zeta _2\in \mathbb {K}$ and $\xi \in \mathbb {K}^n$ , we define the multiplier

$$ \begin{align*} m_N(\zeta_1, \zeta_2, \xi):=\int_{B_1(0)} \mathrm{e}\Big(-\zeta_1P_1(\alpha y)+\zeta_2 P_2(\alpha y)+\sum_{j=1}^n\xi_jQ_j(\alpha y)\Big) d\mu(y), \end{align*} $$

where $\alpha \in \mathbb {K}$ satisfies $|\alpha | = N$ . By definitions (6.45) and (6.46) and making the change of variables $x\mapsto x-P_2(y)$ , we may write

$$ \begin{align*} N_0^{-1}\widehat{F_2^{\xi}}(\zeta_2)&=N_0^{-1}\int_{\mathbb{K}}\int_{\mathbb{K}}F_{2;y}^{\xi}(x)\mathrm{e}(-\zeta_2 x)d\mu_{[N]}(y)d\mu(x)\\ &=N_0^{-1}\int_{\mathbb{K}}\widehat{f}_0(\zeta_2-\zeta_1)\widehat{f}_1(\zeta_1)m_N(\zeta_1, \zeta_2, \xi)d\zeta_1. \end{align*} $$

By the Cauchy–Schwarz inequality and Plancherel’s theorem, we obtain

$$ \begin{align*} \delta\le N_0^{-1}|\widehat{F_2^{\xi}}(\zeta_2)|\lesssim N_0^{-1}\|f_0\|_{L^2(\mathbb{K})} \|f_1\|_{L^2(\mathbb{K})}\sup_{\zeta_1\in \mathbb{K}}|m_N(\zeta_1, \zeta_2, \xi)|, \end{align*} $$

which gives for some $\zeta _1\in \mathbb {K}$ that

$$ \begin{align*} \delta\lesssim |m_N(\zeta_1, \zeta_2, \xi)| \end{align*} $$

since $\|f_0\|_{L^2(\mathbb {K})}, \|f_1\|_{L^2(\mathbb {K})}\lesssim N_0^{1/2}$ . Applying Lemma 6.49 with ${\mathcal P}=\{-P_1, P_2\}$ and $\mathcal Q=\{Q_1,\ldots , Q_n\}$ , we deduce that for every sufficiently large $C\gtrsim 1$ one has

This completes the proof of Claim 6.73 for $m=2$ .

Step 2.

In this step, we show that Claim 6.73 for all integers $m\ge 2$ implies Theorem 6.57. In view of Step 1, this will in particular establish Theorem 6.57 for $m=2$ , which is the base case of our double induction. As before, we shall write ${\boldsymbol{\nu }}_{j}:=\bigotimes _{i=1}^{j}\nu _{[H_i]}$ for any $j\in \mathbb Z_+$ . Recall that $N_0 = N^{\deg (P_m)}$ and note

$$ \begin{align*} \|F_m^{\xi}\|_{\square_{[H_1], \ldots, [H_s]}^s(I)}^{2^s}= \int_{\mathbb{K}^{s-2}}\|{\Delta}_h^{s-2}F_m^{\xi}\|_{\square_{[H_{s-1}], [H_s]}^2(I)}^4 d{\boldsymbol{\nu }}_{s-2}(h). \end{align*} $$

By Equation (6.58) and the pigeonhole principle, there exists a measurable set $X\subseteq \prod _{i=1}^{s-2}[H_i]$ so that ${\boldsymbol{\nu }}_{s-2}(X)\gtrsim \delta ^{O(1)}$ , and for all $h \in X$ one has

$$ \begin{align*} \|{\Delta}_h^{s-2}F_m^{\xi}\|_{\square_{[H_{s-1}], [H_s]}^2(I)} \gtrsim\delta^{O(1)}. \end{align*} $$

Here, we used that $\operatorname {\mathrm {supp}} F_m^{\xi }$ is a subset of an interval whose measure is at most $O(N_0)$ . By Lemma 5.1, we have

Next, we claim that there is a countable set ${\mathcal F} \subset \mathbb {K}$ , depending on N and $\delta $ such that

(6.76)

for some absolute constant $C_0\in \mathbb Z_+$ and for all $h\in X$ . When $\mathbb {K}$ is non-Archimedean, we take

$$ \begin{align*}{\mathcal F} \ = \ \bigcup_{M\ge 1} \, \Big\{ z = \sum_{j= -M}^{L-1} z_j \pi^j \in \mathbb{K} : z_j \in o_{\mathbb{K}}/m_{\mathbb{K}}\Big\}, \end{align*} $$

where $N_0 = q^L$ . Let $x \in I = B_{N_0}(x_0)$ . For any $\zeta \in \mathbb {K}$ , we have $\zeta \in B_{N_0^{-1}}(\zeta _0)$ for some $\zeta _0 \in {\mathcal F}$ . Note that

$$ \begin{align*}\mathrm{e}(-\zeta x) \ = \ \mathrm{e}(-x \zeta_0) \, \mathrm{e}(-(x-x_0)(\zeta - \zeta_0)) \, \mathrm{e}(-x_0(\zeta - \zeta_0)) \ = \ \mathrm{e}(-x \zeta_0) \, \mathrm{e}(-x_0(\zeta - \zeta_0)) \end{align*} $$

since $|(x-x_0)(\zeta - \zeta _0)| \le N_0 N_0^{-1} = 1$ and $\mathrm {e} = 1$ on $o_{\mathbb {K}}$ . Therefore, since ${\Delta }_h^{s-2}F_m$ is supported in I whenever $h \in X$ . This shows that Equation (6.76) holds for non-Archimedean fields.

When $\mathbb {K}= {\mathbb R}$ , we take ${\mathcal F} := T_0 {\mathbb Z}$ , where

$$ \begin{align*} T_0:=\delta^{C_0}\big(CN_0\big)^{-1} \end{align*} $$

for a sufficiently large constant $C\gtrsim 1$ . When $\mathbb {K} = {\mathbb C}$ , we take ${\mathcal F} := T_1 {\mathbb Z}^2$ , where $T_1 := \delta ^{C_0} (C \sqrt {N_0})^{-1}$ . By the Lipschitz nature of characters on ${\mathbb R}$ or ${\mathbb C}$ , we again see that Equation (6.76) holds in the Archimedean cases. In particular, there exists a measurable function $\phi : X \to {\mathcal F}$ so that

(6.77)

for all $h\in X$ . If necessary, we may additionally assume that the range of $\phi $ is finite.

By Lemma 6.66, it follows that

$$ \begin{align*} \int_{\square_{s-2}(X)}\bigg|N^{-1}_0\int_{\mathbb{K}} F_m(x; h, h')e(-\psi((h, h'))x)d\mu(x)\bigg|^2 d{\boldsymbol{\nu }}_{s-2}^{\otimes2}(h, h') \gtrsim \delta^{O(1)}, \end{align*} $$

where

$$ \begin{align*} F_m(x; h, h')&:=\int_{\mathbb{K}}{\Delta}_{h'-h}^{s-2}f_0(x+P_m(y))\prod_{i=1}^{m-1}{\Delta}_{h'-h}^{s-2}f_{i}(x-P_i(y)+P_m(y))d\mu_{[N]}(y),\\\psi(h, h')&:=\sum_{\omega\in\{0, 1\}^{s-2}}(-1)^{|\omega|}\phi\Big(\omega\cdot h+(1-\omega)\cdot h')\Big). \end{align*} $$

Thus, by the pigeonhole principle, there exists a measurable set $X_0\subseteq \square _{s-2}(X)$ with ${\boldsymbol{\nu }}_{s-2}^{\otimes 2}(X_0)\gtrsim \delta ^{O(1)}$ such that for every $(h, h')\in X_0$ one has

$$ \begin{align*} \bigg|N^{-\deg(P_m)}\int_{\mathbb{K}} F_m(x; h, h')e(-\psi((h, h'))x)d\mu(x)\bigg| \gtrsim \delta^{O(1)}. \end{align*} $$

By Claim 6.73, there is a $c:=c_{m, s}\ge 1$ such that for each $(h, h')\in X_0$ , one has

$$ \begin{align*} |\psi((h, h'))|\lesssim_{m, s} \delta^{-c} N^{-\deg(P_m)}. \end{align*} $$

By the pigeonhole principle, there exists $h'\in \prod _{i=1}^{s-2}[H_i]$ and a measurable set

$$\begin{align*}X_0(h'):=\big\{h\in X: (h, h')\in X_0\ \text{ and } \ |\psi((h, h'))|\lesssim \delta^{-c} N^{-\deg(P_m)}\big\} \end{align*}$$

satisfying ${\boldsymbol{\nu }}_{s-2}(X_0(h'))\gtrsim \delta ^{O(1)}$ . Since $\psi ((h, h'))\in {\mathcal F}$ , we see that

$$ \begin{align*} X_0(h')\subseteq\bigcup_{k\in {\mathbb{K}}}X_{0}^k(h'), \end{align*} $$

where ${\mathbb {K}} = [{O(\delta ^{-O(1)})}]\cap {\mathbb Z}$ when $\mathbb {K} = {\mathbb R}$ . In this case, $X_{0}^k(h'):=\{h\in X: \psi ((h, h'))=T_0k\}$ . When $\mathbb {K} = {\mathbb C}$ , we have ${\mathbb {K}} = [{O(\delta ^{-O(1)})}]\cap {\mathbb Z}^2$ and $X_{0}^k(h'):=\{h\in X: \psi ((h, h'))=T_1 k\}$ . Finally when $\mathbb {K}$ is non-Archimedean,

$$ \begin{align*}{\mathbb{K}} = \bigl[{O(\delta^{-O(1)})}\bigr] \cap \bigl\{ k = \sum_{j= -M}^{-1} k_j \pi^j \in \mathbb{K} : k_j \in o_{\mathbb{K}}/m_{\mathbb{K}}\bigr\} \end{align*} $$

and $X_{0}^k(h') := \{h\in X: \psi ((h,h')) = \pi ^L k \}$ .

Thus, by the pigeonhole principle there is $k_0\in {\mathbb {K}}$ such that ${\boldsymbol{\nu }}_{s-2}(X_0^{k_0}(h'))\big )\gtrsim \delta ^{O(1)}$ . When $\mathbb {K} = {\mathbb R}$ , this shows that $\psi (h, h')=T_0 k_0 =: \phi _m$ for all $h\in X_0^{k_0}(h')$ . When $\mathbb {K} = {\mathbb C}$ , we have $\phi (h,h') = T_1 k_0$ for all $h \in X_0^{k_0}(h')$ and when $\mathbb {K}$ is non-Archimedean, $\phi (h,h') = \pi ^L k_0$ for all $h \in X_0^{k_0}(h')$ . We will denote these values by $\phi _m$ in all cases.

Set

$$ \begin{align*} \psi_1(h, h'):=(-1)^{s+1}\sum_{\substack{\omega\in\{0,1\}^{s-2} \\ \omega_1=0}}(-1)^{|\omega|}\phi\Big((\omega\cdot h+(1-\omega)\cdot h')\Big) + (-1)^s \phi_m \end{align*} $$

and, for , set

$$ \begin{align*} \psi_i(h, h'):=(-1)^{s+1}\sum_{\substack{\omega\in\{0,1\}^{s-2}\setminus\{0\} \\ \omega_1=\ldots=\omega_{i-1}=1 \\ \omega_{i}=0}}(-1)^{|\omega|}\phi\Big((\omega\cdot h+(1-\omega)\cdot h')\Big). \end{align*} $$

Note that $\psi _i$ does not depend on $h_{i}$ and we can write