1 Introduction
Let
$A=(a_{i,j})_{1\leqslant i,j\leqslant n}$
be a symmetric integral matrix with
$n\geqslant 4$
. In other words,
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}a_{1,1} & \cdots \, & a_{1,n}\\ \vdots & \cdots \, & \vdots \\ a_{n,1} & \cdots \, & a_{n,n}\end{array}\right)\end{eqnarray}$$
with
$a_{i,j}=a_{j,i}\in \mathbb{Z}$
for all
$1\leqslant i<j\leqslant n$
. Let
$f(x_{1},\ldots ,x_{n})$
be the quadratic form defined as
$$\begin{eqnarray}f(x_{1},\ldots ,x_{n})=\mathop{\sum }_{i=1}^{n}\mathop{\sum }_{j=1}^{n}a_{i,j}x_{i}x_{j}.\end{eqnarray}$$
Let
$t$
be an integer. We call
$f$
regular if
$A$
is invertible. For regular indefinite quadratic forms with
$n\geqslant 4$
, the well-known Hasse principle asserts that
$f(x_{1},\ldots ,x_{n})=t$
has integer solutions if and only if
$f(x_{1},\ldots ,x_{n})=t$
has local solutions.
In this paper, we consider the equation
$f(x_{1},\ldots ,x_{n})=t$
, where
$x_{1},\ldots ,x_{n}$
are prime variables. It is expected that
$f(x_{1},\ldots ,x_{n})=t$
has solutions with
$x_{1},\ldots ,x_{n}$
primes if there are suitable local solutions. The classical theorem of Hua [Reference Hua7] deals with diagonal quadratic forms in five prime variables. In particular, every sufficiently large integer, congruent to 5 modulo 24, can be represented as a sum of five squares of primes. Recently, Liu [Reference Liu9] handled a wide class of quadratic forms
$f$
with 10 or more prime variables. The general quadratic form in prime variables (or in dense sets) was recently investigated by Cook and Magyar [Reference Cook and Magyar3], and by Keil [Reference Keil8]. In particular, Cook and Magyar [Reference Cook and Magyar3] handled all regular quadratic forms in 21 or more prime variables, while the work of Keil [Reference Keil8] can deal with all regular quadratic forms in 17 or more variables. It involves only five prime variables for diagonal quadratic equation due to the effective mean value theorem. This is similar to the problem concerning Diophantine equations for cubic forms. The works of Baker [Reference Baker1], Vaughan [Reference Vaughan10, Reference Vaughan11] and Wooley [Reference Wooley13, Reference Wooley14] can deal with the diagonal cubic equation in seven variables. However, more variables are involved for general cubic forms. One can refer to the works of Heath-Brown [Reference Heath-Brown4, Reference Heath-Brown5] and Hooley [Reference Hooley6] for general cubic forms.
The purpose of this paper is to investigate general regular quadratic forms in nine or more prime variables. We define
$$\begin{eqnarray}N_{f,t}(X)=\mathop{\sum }_{\substack{ 1\leqslant x_{1},\ldots ,x_{n}\leqslant X \\ f(x_{1},\ldots ,x_{n})=t}}\mathop{\prod }_{j=1}^{n}{\rm\Lambda}(x_{j}),\end{eqnarray}$$
where
${\rm\Lambda}(\cdot )$
is the von Mangoldt function. Our main result is the following.
Theorem 1.1. Suppose that
$f(x_{1},\ldots ,x_{n})$
is a regular integral quadratic form with
$n\geqslant 9$
, and that
$t\in \mathbb{Z}$
. Let
$\mathfrak{S}(f,t)$
and
$\mathfrak{I}_{f,t}(X)$
be defined in (3.11) and (3.13), respectively. Suppose that
$K$
is an arbitrary large real number. Then we have
$$\begin{eqnarray}N_{f,t}(X)=\mathfrak{S}(f,t)\mathfrak{I}_{f,t}(X)+O(X^{n-2}\log ^{-K}X),\end{eqnarray}$$
where the implied constant depends on
$f$
and
$K$
.
Denote by
$\mathbb{P}$
the set of all prime numbers. For a prime
$p\in \mathbb{P}$
, we use
$\mathbb{Z}_{p}$
to denote the ring of
$p$
-adic integers. Then we use
$\mathbb{U}_{p}$
to denote the set of
$p$
-adic units in
$\mathbb{Z}_{p}$
. The general local to global conjecture of Bourgain–Gamburd–Sarnak [Reference Bourgain, Gamburd and Sarnak2] asserts that
$f(x_{1},\ldots ,x_{n})=t$
has prime solutions provided that there are local solutions in
$\mathbb{U}_{p}$
for all
$p\in \mathbb{P}$
. Liu [Reference Liu9, Theorem 1.1] verified this conjecture for a wide class of regular indefinite integral quadratic forms with ten or more variables. Theorem 1.1 has the following consequence improving upon Liu [Reference Liu9, Theorem 1.1].
Theorem 1.2. Let
$f(x_{1},\ldots ,x_{n})$
be a regular indefinite integral quadratic form with
$n\geqslant 9$
, and let
$t\in \mathbb{Z}$
. Then
$f(x_{1},\ldots ,x_{n})=t$
has prime solutions if we have the following two conditions:
-
(i) there are real solutions in
$\mathbb{R}^{+}$
, and -
(ii) there are local solutions in
$\mathbb{U}_{p}$
for all prime
$p$
.
We define
$N_{f,t}^{\ast }(X)$
to be the number of prime solutions to
$f(p_{1},\ldots ,p_{n})=t$
with
$1\leqslant p_{1},\ldots ,p_{n}\leqslant X$
. Suppose that
$f$
is regular with
$n\geqslant 9$
. Actually, in view of Theorem 1.1, one has
$N_{f,t}^{\ast }(X)\gg _{f,t}X^{n-2}\log ^{-n}X$
for sufficiently large
$X$
if the conditions (i) and (ii) in Theorem 1.2 hold.
Theorem 1.2 covers all regular indefinite integral quadratic forms in nine prime variables. The
$O$
-constant in the asymptotic formula (1.3) is independent of
$t$
. Therefore, Theorem 1.1 can be applied to definite quadratic forms. In particular, if
$f(x_{1},\ldots ,x_{n})$
is a positive definite integral quadratic form with
$n\geqslant 9$
, then there exist
$r,q\in \mathbb{N}$
so that all sufficiently large natural numbers
$N$
, congruent to
$r$
modulo
$q$
, can be represented as
$N=f(p_{1},\ldots ,p_{n})$
, where
$p_{1},\ldots ,p_{n}$
are prime numbers.
The method in this paper can also be applied to refine Keil [Reference Keil8, Theorem 1.1]. In particular, one may obtain a variant of Keil [Reference Keil8, Theorem 1.1] for a wide class of quadratic forms in nine variables.
2 Notations
As usual, we write
$e(z)$
for
$e^{2{\it\pi}iz}$
. Throughout we assume that
$X$
is sufficiently large. Let
$L=\log X$
. We use
$\ll$
and
$\gg$
to denote Vinogradov’s well-known notations, while the implied constants may depend on the form
$f$
. Denote by
${\it\phi}(q)$
Euler’s totient function.
For a set
${\mathcal{S}}$
in a field
$\mathbb{F}$
, we define
$$\begin{eqnarray}{\mathcal{S}}^{n}=\{(x_{1},\ldots ,x_{n})^{T}:~x_{1},\ldots ,x_{n}\in {\mathcal{S}}\}.\end{eqnarray}$$
We use
$M_{m,n}({\mathcal{S}})$
to denote the set of
$m$
by
$n$
matrixes
$$\begin{eqnarray}M_{m,n}({\mathcal{S}})=\{(a_{i,j})_{1\leqslant i\leqslant m,\,1\leqslant j\leqslant n}:~a_{i,j}\in {\mathcal{S}}\},\end{eqnarray}$$
and
$GL_{n}({\mathcal{S}})$
to denote the set of invertible matrixes of order
$n$
$$\begin{eqnarray}GL_{n}({\mathcal{S}})=\{B\in M_{n,n}({\mathcal{S}}):~B\text{ is invertible}\},\end{eqnarray}$$
respectively. We define the off-diagonal rank of
$A$
as
$$\begin{eqnarray}\text{rank}_{\text{off}}(A)=\max \{r:~r\in R\},\end{eqnarray}$$
where
$$\begin{eqnarray}R=\{\text{rank}(B):~B=(a_{i_{k},j_{l}})_{1\leqslant k,l\leqslant r}\text{ with }\{i_{1},\ldots ,i_{r}\}\cap \{j_{1},\ldots ,j_{r}\}=\emptyset \}.\end{eqnarray}$$
In other words,
$\text{rank}_{\text{off}}(A)$
is the maximal rank of a submatrix in
$A$
, which does not contain any diagonal entries. For
$\mathbf{x}=(x_{1},\ldots ,x_{n})^{T}\in \mathbb{N}^{n}$
, we write
$$\begin{eqnarray}{\rm\Lambda}(\mathbf{x})={\rm\Lambda}(x_{1})\cdots {\rm\Lambda}(x_{n}).\end{eqnarray}$$
For
$\mathbf{x}=(x_{1},\ldots ,x_{n})^{T}\in \mathbb{Z}^{n}$
, we also use the notation
${\mathcal{A}}(\mathbf{x})$
to indicate that the argument
${\mathcal{A}}(x_{j})$
holds for all
$1\leqslant j\leqslant s$
. The meaning will be clear from the text. For example, we use
$1\leqslant \mathbf{x}\leqslant X$
and
$|\mathbf{x}|\leqslant X$
to denote
$1\leqslant x_{j}\leqslant X$
for
$1\leqslant j\leqslant n$
and
$|x_{j}|\leqslant X$
for
$1\leqslant j\leqslant n$
, respectively.
In order to apply the circle method, we introduce the exponential sum
$$\begin{eqnarray}S({\it\alpha})=\mathop{\sum }_{1\leqslant \mathbf{x}\leqslant X}{\rm\Lambda}(\mathbf{x})e({\it\alpha}\mathbf{x}^{T}A\mathbf{x}),\end{eqnarray}$$
where
$A$
is defined in (1.1). We define
$$\begin{eqnarray}{\mathcal{M}}(Q)=\mathop{\bigcup }_{1\leqslant q\leqslant Q}\mathop{\bigcup }_{\substack{ a=1 \\ (a,q)=1}}^{q}{\mathcal{M}}(q,a;Q),\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{M}}(q,a;Q)=\biggl\{{\it\alpha}:~\bigg|{\it\alpha}-\frac{a}{q}\bigg|\leqslant \frac{Q}{qX^{2}}\biggr\}.\end{eqnarray}$$
The intervals
${\mathcal{M}}(q,a;Q)$
are pairwise disjoint for
$1\leqslant a\leqslant q\leqslant Q$
and
$(a,q)=1$
provided that
$Q\leqslant X/2$
. For
$Q\leqslant X/2$
, we set
$$\begin{eqnarray}\mathfrak{m}(Q)={\mathcal{M}}(2Q)\setminus {\mathcal{M}}(Q).\end{eqnarray}$$
Now we introduce the major arcs defined as
$$\begin{eqnarray}\mathfrak{M}={\mathcal{M}}(P)\quad \text{with}\;P=L^{K},\end{eqnarray}$$
where
$K$
is a sufficiently large constant throughout this paper. Then we define the minor arcs as
$$\begin{eqnarray}\mathfrak{m}=[X^{-1},1+X^{-1}]\setminus \mathfrak{M}.\end{eqnarray}$$
3 The contribution from the major arcs
For
$q\in \mathbb{N}$
and
$(a,q)=1$
, we define
$$\begin{eqnarray}C(q,a)=\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant q \\ (\mathbf{h},q)=1}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{q}\bigg),\end{eqnarray}$$
where
$A$
is given by (1.1). Throughout, we assume that
$f$
is connected to
$A$
given by (1.1) and (1.2). Let
$$\begin{eqnarray}B_{f,t}(q)=\frac{1}{{\it\phi}^{n}(q)}\mathop{\sum }_{\substack{ a=1 \\ (a,q)=1}}^{q}C(q,a)e\bigg(-\frac{at}{q}\bigg).\end{eqnarray}$$
Concerning
$B_{f,t}(q)$
, we have the following multiplicative property.
Lemma 3.1. The arithmetic function
$B_{f,t}(q)$
is multiplicative.
Proof. The desired conclusion can be proved by changing variables. ◻
Lemma 3.2. Suppose that
$A$
is invertible. For any prime
$p$
, there exists
${\it\gamma}_{p}={\it\gamma}_{p}(f,t)$
such that
$B_{f,t}(p^{k})=0$
for all
$k>{\it\gamma}_{p}$
. Moreover, if
$p\nmid 2\det (A)$
, then we have
${\it\gamma}_{p}=1$
.
Proof. Throughout this proof, we assume that
$(a,p)=1$
. We first deal with the case
$p\geqslant 3$
. We claim that if
$$\begin{eqnarray}C(p^{k},a)=p^{nj}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-j} \\ (\mathbf{h},p)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{j})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg)\end{eqnarray}$$
for some
$j\leqslant (k-2)/2$
, then
$$\begin{eqnarray}C(p^{k},a)=p^{n(j+1)}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-j-1} \\ (\mathbf{h},p)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{j+1})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg).\end{eqnarray}$$
Indeed, by changing variables, we obtain from (3.3) that
$$\begin{eqnarray}\displaystyle C(p^{k},a) & = & \displaystyle p^{nj}\mathop{\sum }_{1\leqslant \mathbf{u}\leqslant p}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-j-1} \\ (\mathbf{h},p)=1 \\ A(\mathbf{u}p^{k-j-1}+\mathbf{h})\equiv \mathbf{0}(\text{mod}~p^{j})}}\nonumber\\ \displaystyle & & \displaystyle \times \,e\bigg((\mathbf{u}p^{k-j-1}+\mathbf{h})^{T}A(\mathbf{u}p^{k-j-1}+\mathbf{h})\frac{a}{p^{k}}\bigg).\nonumber\end{eqnarray}$$
It follows from
$j\leqslant (k-2)/2$
that
$j\leqslant k-j-1$
and
$k\leqslant 2(k-j-1)$
. Thus we deduce that
$$\begin{eqnarray}\displaystyle C(p^{k},a) & = & \displaystyle p^{nj}\mathop{\sum }_{1\leqslant \mathbf{u}\leqslant p}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-j-1} \\ (\mathbf{h},p)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{j})}}e\bigg(2p^{k-j-1}\mathbf{u}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg)e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg)\nonumber\\ \displaystyle & = & \displaystyle p^{n(j+1)}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-j-1} \\ (\mathbf{h},p)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{j+1})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg).\nonumber\end{eqnarray}$$
This establishes the desired claim, and therefore we arrive at
$$\begin{eqnarray}C(p^{k},a)=p^{ns}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{k-s} \\ (\mathbf{h},p)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{s})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p^{k}}\bigg),\end{eqnarray}$$
where
$s=\lfloor k/2\rfloor$
. There exists
$P\in GL_{n}(\mathbb{Z}_{p})$
with
$\det (P)=1$
such that
$P^{T}AP=D=\text{diag}\{d_{1},\ldots ,d_{n}\}$
with
$d_{1},\ldots ,d_{n}\in \mathbb{Z}_{p}$
. Note that
$A$
is invertible, one has
$d_{1}\cdots d_{n}\not =0$
. In particular, we can choose
$r\in \mathbb{N}$
such that
$p^{r}\nmid d_{j}$
for all
$1\leqslant j\leqslant n$
. The condition
$A\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{s})$
implies
$DP\mathbf{h}\equiv \mathbf{0}(\text{mod}~p^{s})$
. If
$s\geqslant r$
, then
$P\mathbf{h}\equiv \mathbf{0}(\text{mod}~p)$
. So we obtain
$\mathbf{h}\equiv \mathbf{0}(\text{mod}~p)$
, which is a contradiction to the condition
$(\mathbf{h},p)=1$
. Therefore, we conclude that
$$\begin{eqnarray}C(p^{k},a)=0\quad \text{for all }k\geqslant 2r.\end{eqnarray}$$
Moreover, when
$p\nmid 2\det (A)$
, we can take
$r=1$
in (3.6).
For
$p=2$
, the above argument is still valid with minor modifications. We now claim that if
$$\begin{eqnarray}C(2^{k},a)=2^{2nj}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant 2^{k-2j} \\ (\mathbf{h},2)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~2^{j})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{2^{k}}\bigg)\end{eqnarray}$$
for some
$j\leqslant (k-4)/4$
, then
$$\begin{eqnarray}C(2^{k},a)=2^{2n(j+1)}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant 2^{k-2j-2} \\ (\mathbf{h},2)=1 \\ A\mathbf{h}\equiv \mathbf{0}(\text{mod}~2^{j+1})}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{2^{k}}\bigg).\end{eqnarray}$$
This claim can be established by changing variables
$\mathbf{h}=\mathbf{u}2^{k-2j-2}+\mathbf{v}$
with
$\mathbf{u}(\text{mod}~2^{2})$
and
$\mathbf{v}(\text{mod}~2^{k-2j-2})$
. The argument leading to (3.6) implies that there exists
$k_{0}$
such that
$$\begin{eqnarray}C(2^{k},a)=0\quad \text{for all }k\geqslant k_{0}.\end{eqnarray}$$
The desired conclusion follows from (3.2), (3.6) and (3.9).◻
Lemma 3.3. Let
$B_{f,t}(q)$
be defined as (3.2). If
$A$
is invertible and
$n\geqslant 5$
, then
$$\begin{eqnarray}B_{f,t}(q)\ll _{f,{\it\varepsilon}}q^{-3/2+{\it\varepsilon}}.\end{eqnarray}$$
Proof. In view of Lemma 3.2, it suffices to prove
$$\begin{eqnarray}C(p,a)\ll _{f}p^{n-5/2}\end{eqnarray}$$
for
$p\nmid 2\det (A)$
and
$(a,p)=1$
. Note that
$$\begin{eqnarray}\displaystyle C(p,a) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{N}^{n} \\ 1\leqslant \mathbf{h}\leqslant p}}e\bigg(\mathbf{h}^{T}A\mathbf{h}\frac{a}{p}\bigg)-\mathop{\sum }_{j=1}^{n}\mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{N}^{n-1} \\ 1\leqslant \mathbf{h}\leqslant p}}e\bigg(\mathbf{h}^{T}A_{j}\mathbf{h}\frac{a}{p}\bigg)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{1\leqslant i<j\leqslant n}\mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{N}^{n-2} \\ 1\leqslant \mathbf{h}\leqslant p}}e\bigg(\mathbf{h}^{T}A_{ij}\mathbf{h}\frac{a}{p}\bigg)+O(p^{n-3}),\nonumber\end{eqnarray}$$
where
$A_{j}$
denotes the submatrix of
$A$
by deleting the
$j$
th row and
$j$
th column, and
$A_{ij}$
denotes the submatrix of
$A_{j}$
by deleting the
$i$
th row and
$i$
th column. For complete Gauss sums, we have
$$\begin{eqnarray}\mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{N}^{k} \\ 1\leqslant \mathbf{h}\leqslant p}}e\bigg(\mathbf{h}^{T}M\mathbf{h}\frac{a}{p}\bigg)\ll p^{k-\text{rank}(M)/2},\end{eqnarray}$$
where the implied constant depends on the square matrix
$M$
. The estimate (3.10) follows by observing that
$\text{rank}(A_{j})\geqslant 3$
and
$\text{rank}(A_{ij})\geqslant 1$
. We complete the proof.◻
Now we introduce the singular series
$\mathfrak{S}(f,t)$
defined as
$$\begin{eqnarray}\mathfrak{S}(f,t)=\mathop{\sum }_{q=1}^{\infty }B_{f,t}(q),\end{eqnarray}$$
where
$B_{f,t}(q)$
is given by (3.2). From Lemmas 3.2 and 3.3, we conclude the following result.
Lemma 3.4. Suppose that
$A$
is invertible and
$n\geqslant 5$
. Then the singular series
$\mathfrak{S}(f,t)$
is absolutely convergent, and
$$\begin{eqnarray}\mathfrak{S}(f,t)=\mathop{\prod }_{p}{\it\chi}_{p}(f,t),\end{eqnarray}$$
where the local densities
${\it\chi}_{p}(f,t)$
are defined as
$$\begin{eqnarray}{\it\chi}_{p}(f,t)=1+\mathop{\sum }_{m=1}^{\infty }B_{f,t}(p^{m}).\end{eqnarray}$$
Moreover, if
$f(x_{1},\ldots ,x_{n})=t$
has local solutions in
$\mathbb{U}_{p}$
for all prime
$p$
, then one has
$$\begin{eqnarray}\mathfrak{S}(f,t)\gg 1.\end{eqnarray}$$
Proof. It suffices to explain
$\mathfrak{S}(f,t)\gg 1$
provided that
$f(x_{1},\ldots ,x_{n})=t$
has local solutions in
$\mathbb{U}_{p}$
for all prime
$p$
. Indeed, in view of Lemma 3.3, one has
$\prod _{p\geqslant p_{0}}{\it\chi}_{p}(f,t)\gg 1$
for some
$p_{0}$
. When
$p<p_{0}$
, by Lemma 3.2, for some
${\it\gamma}={\it\gamma}_{p}$
we have
$$\begin{eqnarray}{\it\chi}_{p}(f,t)=1+\mathop{\sum }_{m=1}^{{\it\gamma}}B_{f,t}(p^{m})=\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant p^{{\it\gamma}} \\ (\mathbf{h},p)=1 \\ F(\mathbf{h})\equiv t(\text{mod}~p^{{\it\gamma}})}}1.\end{eqnarray}$$
Since
$f(x_{1},\ldots ,x_{n})=t$
has local solutions in
$\mathbb{U}_{p}$
, one has
${\it\chi}_{p}(f,t)>0$
. This concludes that
$\mathop{\prod }_{p}{\it\chi}_{p}(f,t)\gg 1$
.◻
Remark 3.5. We point out that in view of the proof of Lemmas 3.2–3.3, one has
$$\begin{eqnarray}B_{f,t}(2^{k}q_{1}q_{2})\ll _{f}2^{-(\text{rank}(A)/4)k}q_{1}^{-\text{rank}(A)/2}q_{2}^{-\text{rank}(A)/3},\end{eqnarray}$$
where
$q_{1}$
is square-free and
$(2,q_{1}q_{2})=(q_{1},q_{2})=1$
. In particular, the singular series is absolutely convergent if
$\text{rank}(A)\geqslant 5$
. Therefore, the condition that
$f$
is regular with
$n\geqslant 9$
in our Theorem 1.1 can be replaced by
$\text{rank}(A)\geqslant 9$
.
We define
$$\begin{eqnarray}I({\it\beta})=\int _{[0,X]^{n}}e({\it\beta}\mathbf{x}^{T}A\mathbf{x})d\mathbf{x}.\end{eqnarray}$$
Since
$I({\it\beta})\ll X^{n}(1+X^{2}|{\it\beta}|)^{-2}$
for
$\text{rank}(A)\geqslant 5$
, we introduce the singular integral
$$\begin{eqnarray}\mathfrak{I}_{f,t}(X)=\int _{-\infty }^{\infty }I({\it\beta})e(-t{\it\beta})d{\it\beta},\end{eqnarray}$$
where
$f(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}$
. Note that
$\mathfrak{I}_{f,t}(X)\gg _{f,t}X^{n-2}$
if
$f(x_{1},\ldots ,x_{n})$
is indefinite and
$f(x_{1},\ldots ,x_{n})=t$
has positive real solutions.
Lemma 3.6. Let
$t\in \mathbb{Z}$
, and let
$$\begin{eqnarray}S({\it\alpha})=\mathop{\sum }_{1\leqslant \mathbf{x}\leqslant X}{\rm\Lambda}(\mathbf{x})e({\it\alpha}\mathbf{x}^{T}A\mathbf{x}),\end{eqnarray}$$
where
$A\in M_{n,n}(\mathbb{Z})$
is a symmetric matrix with
$\text{rank}(A)\geqslant 5$
. Then one has
$$\begin{eqnarray}\int _{\mathfrak{M}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}=\mathfrak{S}(f,t)\mathfrak{I}_{f,t}(X)+O(X^{n-2}L^{-K/4}).\end{eqnarray}$$
Proof. We write
$f(\mathbf{x})$
for
$\mathbf{x}^{T}A\mathbf{x}$
. By the definition of
$\mathfrak{M}$
, one has
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathfrak{M}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{q\leqslant P}\mathop{\sum }_{\substack{ 1\leqslant a\leqslant q \\ (a,q)=1}}\int _{|{\it\beta}|\leqslant \frac{P}{qX^{2}}}\mathop{\sum }_{1\leqslant \mathbf{x}\leqslant X}{\rm\Lambda}(\mathbf{x})e\bigg(f(\mathbf{x})\bigg(\frac{a}{q}+{\it\beta}\bigg)\bigg)e\bigg(-t\bigg(\frac{a}{q}+{\it\beta}\bigg)\bigg)d{\it\beta}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
We introduce the congruence condition to deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{1\leqslant \mathbf{x}\leqslant X}{\rm\Lambda}(\mathbf{x})e\bigg(f(\mathbf{x})\bigg(\frac{a}{q}+{\it\beta}\bigg)\bigg)\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{1\leqslant \mathbf{h}\leqslant q}e\bigg(f(\mathbf{h})\frac{a}{q}\bigg)\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X \\ \mathbf{x}\equiv \mathbf{h}(\text{mod}~q)}}{\rm\Lambda}(\mathbf{x})e(f(\mathbf{x}){\it\beta})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ 1\leqslant \mathbf{h}\leqslant q \\ (\mathbf{h},q)=1}}e\bigg(f(\mathbf{h})\frac{a}{q}\bigg)\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X \\ \mathbf{x}\equiv \mathbf{h}(\text{mod}~q)}}{\rm\Lambda}(\mathbf{x})e(f(\mathbf{x}){\it\beta})+O(X^{n-1}LP).\nonumber\end{eqnarray}$$
Since
$q\leqslant P=L^{K}$
, the Siegel–Walfisz theorem together with summation by parts will imply for
$(\mathbf{h},q)=1$
that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X \\ \mathbf{x}\equiv \mathbf{h}(\text{mod}~q)}}{\rm\Lambda}(\mathbf{x})e(f(\mathbf{x}){\it\beta}) & = & \displaystyle \frac{1}{{\it\phi}^{n}(q)}\int _{[0,X]^{n}}e(f(\mathbf{x}){\it\beta})d\mathbf{x}+O(X^{n}L^{-100K})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{{\it\phi}^{n}(q)}I({\it\beta})+O(X^{n}L^{-100K}).\nonumber\end{eqnarray}$$
It follows from above
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant \mathbf{x}\leqslant X}{\rm\Lambda}(\mathbf{x})e\bigg(f(\mathbf{x})\bigg(\frac{a}{q}+{\it\beta}\bigg)\bigg)=\frac{C(q,a)}{{\it\phi}^{n}(q)}I({\it\beta})+O(X^{n}L^{-10K}).\end{eqnarray}$$
By putting (3.16) into (3.15), we obtain
$$\begin{eqnarray}\int _{\mathfrak{M}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}=\mathop{\sum }_{q\leqslant P}B_{f,t}(q)\int _{|{\it\beta}|\leqslant \frac{P}{qX^{2}}}I({\it\beta})e(-t{\it\beta})d{\it\beta}+O(X^{n}L^{-K}).\end{eqnarray}$$
It follows from
$I({\it\beta})\ll X^{n}(1+X^{2}|{\it\beta}|)^{-2}$
that
$$\begin{eqnarray}\mathfrak{I}_{f,t}(X)\ll X^{n-2}\end{eqnarray}$$
and
$$\begin{eqnarray}\int _{|{\it\beta}|\leqslant \frac{P}{qX^{2}}}I({\it\beta})e(-t{\it\beta})d{\it\beta}=\mathfrak{I}_{f,t}(X)+O(qX^{n-2}P^{-1}).\end{eqnarray}$$
Combining (3.17)–(3.19) together with Remark 3.5, we conclude
$$\begin{eqnarray}\int _{\mathfrak{M}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}=\mathfrak{S}(f,t)\mathfrak{I}_{f,t}(X)+O(X^{n-2}L^{-K/4}).\end{eqnarray}$$
The proof of Lemma 3.6 is complete. ◻
4 Estimates for exponential sums
Lemma 4.1. Let
$\{{\it\xi}_{z}\}$
be a sequence satisfying
$|{\it\xi}_{z}|\leqslant 1$
. Then one has
$$\begin{eqnarray}\mathop{\sum }_{|y|\ll X}\bigg|\mathop{\sum }_{|z|\ll X}{\it\xi}_{z}e({\it\alpha}yz)\bigg|^{2}\ll X\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\alpha}\Vert ^{-1}\}.\end{eqnarray}$$
Proof. We expand the square to deduce that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{|y|\ll X}\bigg|\mathop{\sum }_{|z|\ll X}{\it\xi}_{z}e({\it\alpha}yz)\bigg|^{2} & = & \displaystyle \mathop{\sum }_{|z_{1}|\ll X}\mathop{\sum }_{|z_{2}|\ll X}{\it\xi}_{z_{1}}\overline{{\it\xi}_{z_{2}}}\mathop{\sum }_{|y|\ll X}e({\it\alpha}y(z_{1}-z_{2}))\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{|z_{1}|\ll X}\mathop{\sum }_{|z_{2}|\ll X}\bigg|\mathop{\sum }_{|y|\ll X}e({\it\alpha}y(z_{1}-z_{2}))\bigg|.\nonumber\end{eqnarray}$$
By changing variables, one can obtain
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{|y|\ll X}\bigg|\mathop{\sum }_{|z|\ll X}{\it\xi}_{z}e({\it\alpha}yz)\bigg|^{2} & \ll & \displaystyle \mathop{\sum }_{|z|\ll X}\mathop{\sum }_{|x|\ll X}\bigg|\mathop{\sum }_{|y|\ll X}e({\it\alpha}yx)\bigg|\nonumber\\ \displaystyle & \ll & \displaystyle X\mathop{\sum }_{|x|\ll X}\bigg|\mathop{\sum }_{|y|\ll X}e({\it\alpha}yx)\bigg|\nonumber\\ \displaystyle & \ll & \displaystyle X\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\alpha}\Vert ^{-1}\}.\nonumber\end{eqnarray}$$
We complete the proof.◻
Lemma 4.2. For
${\it\alpha}\in \mathfrak{m}(Q)$
, one has
$$\begin{eqnarray}\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\alpha}\Vert ^{-1}\}\ll LQ^{-1}X^{2}.\end{eqnarray}$$
Proof. For
${\it\alpha}\in \mathfrak{m}(Q)$
, there exist
$a$
and
$q$
such that
$1\leqslant a\leqslant q\leqslant 2Q$
,
$(a,q)=1$
and
$|{\it\alpha}-a/q|\leqslant 2Q(qX^{2})^{-1}$
. By a variant of Vaughan [Reference Vaughan12, Lemma 2.2] (see also Exercise 2 in Chapter 2 [Reference Vaughan12]), one has
$$\begin{eqnarray}\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\alpha}\Vert ^{-1}\}\ll LX^{2}\bigg(\frac{1}{q(1+X^{2}|{\it\beta}|)}+\frac{1}{X}+\frac{q(1+X^{2}|{\it\beta}|)}{X^{2}}\bigg).\end{eqnarray}$$
Since
${\it\alpha}\in \mathfrak{m}(Q)$
, one has either
$q>Q$
or
$|{\it\alpha}-a/q|>Q(qX^{2})^{-1}$
. Then the desired estimate follows immediately.◻
Lemma 4.3. Let
${\it\alpha}\in \mathfrak{m}$
and
${\it\beta}\in \mathbb{R}$
. For
$d\in \mathbb{Q}$
, we define
$$\begin{eqnarray}f({\it\alpha},{\it\beta})=\mathop{\sum }_{1\leqslant x\leqslant X}{\rm\Lambda}(x)e({\it\alpha}dx^{2}+x{\it\beta}).\end{eqnarray}$$
If
$d\not =0$
, then one has
$$\begin{eqnarray}f({\it\alpha},{\it\beta})\ll XL^{-K/5},\end{eqnarray}$$
where the implied constant depends only on
$d$
and
$K$
.
Proof. The result is essentially classical. In particular, the method used to handle
$\sum _{1\leqslant x\leqslant X}{\rm\Lambda}(x)e({\it\alpha}x^{2})$
can be modified to establish the desired conclusion. We only explain that the implied constant is independent of
${\it\beta}$
. By Vaughan’s identity, we essentially consider two types of exponential sums
$$\begin{eqnarray}\mathop{\sum }_{y}{\it\eta}_{y}\mathop{\sum }_{x}e({\it\alpha}dx^{2}y^{2}+xy{\it\beta})\end{eqnarray}$$
and
$$\begin{eqnarray}\mathop{\sum }_{x}\mathop{\sum }_{y}{\it\xi}_{x}{\it\eta}_{y}e({\it\alpha}dx^{2}y^{2}+xy{\it\beta}).\end{eqnarray}$$
By Cauchy’s inequality, to handle the summation (4.3), it suffices to deal with
$$\begin{eqnarray}\mathop{\sum }_{y_{1}}\mathop{\sum }_{y_{2}}{\it\eta}_{y_{1}}\overline{{\it\eta}_{y_{2}}}\mathop{\sum }_{x}e({\it\alpha}dx^{2}(y_{1}^{2}-y_{2}^{2})+x(y_{1}-y_{2}){\it\beta}).\end{eqnarray}$$
One can apply the differencing argument to the summation of the type
$\sum _{x}e({\it\alpha}^{\prime }x^{2}+x{\it\beta}^{\prime })$
as follows
$$\begin{eqnarray}\displaystyle \bigg|\mathop{\sum }_{x}e({\it\alpha}^{\prime }x^{2}+x{\it\beta}^{\prime })\bigg|^{2} & = & \displaystyle \mathop{\sum }_{x_{1}}\mathop{\sum }_{x_{2}}e({\it\alpha}^{\prime }(x_{1}^{2}-x_{2}^{2})+(x_{1}-x_{2}){\it\beta}^{\prime })\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{h}\mathop{\sum }_{x}e(2{\it\alpha}^{\prime }hx+h{\it\beta}^{\prime })\leqslant \mathop{\sum }_{h}\bigg|\mathop{\sum }_{x}e(2{\it\alpha}^{\prime }hx)\bigg|.\nonumber\end{eqnarray}$$
This leads to the fact that the estimate (4.1) is uniformly for
${\it\beta}$
.◻
Lemma 4.4. Let
${\it\alpha}\in \mathfrak{m}(Q)$
. Suppose that
$A$
is in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & B & 0\\ B^{T} & A_{2} & C\\ 0 & C^{T} & A_{3}\end{array}\right),\end{eqnarray}$$
where
$\text{rank}(B)\geqslant 3$
and
$\text{rank}(C)\geqslant 2$
. Then we have
$$\begin{eqnarray}S({\it\alpha})\ll X^{n}Q^{-5/2}L^{n+5/2}.\end{eqnarray}$$
Remark 4.5. In view of the proof, the estimate (4.5) still holds provided that
$\text{rank}(B)+\text{rank}(C)\geqslant 5$
.
Proof. By (4.4), we can write
$S({\it\alpha})$
in the form
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X \\ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z})\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}+\mathbf{y}^{T}A_{2}\mathbf{y}+2\mathbf{y}^{T}C\mathbf{z}+\mathbf{z}^{T}A_{3}\mathbf{z})),\nonumber\end{eqnarray}$$
where
$\mathbf{x}\in \mathbb{N}^{r}$
,
$\mathbf{y}\in \mathbb{N}^{s}$
and
$\mathbf{z}\in \mathbb{N}^{t}$
. Then we have
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & {\leqslant} & \displaystyle L^{s}\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X}}{\rm\Lambda}(\mathbf{x})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}))\bigg|\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{z})e({\it\alpha}(2\mathbf{y}^{T}C\mathbf{z}+\mathbf{z}^{T}A_{3}\mathbf{z}))\bigg|.\nonumber\end{eqnarray}$$
By Cauchy’s inequality, we obtain
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & {\leqslant} & \displaystyle L^{s}\bigg(\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X}}{\rm\Lambda}(\mathbf{x})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}))\bigg|^{2}\bigg)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg(\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{z})e({\it\alpha}(2\mathbf{y}^{T}C\mathbf{z}+\mathbf{z}^{T}A_{3}\mathbf{z}))\bigg|^{2}\bigg)^{1/2}.\end{eqnarray}$$
We deduce by expanding the square that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X}}{\rm\Lambda}(\mathbf{x})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}))\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}_{1}\leqslant X}}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}_{2}\leqslant X}}{\it\xi}(\mathbf{x}_{1},\mathbf{x}_{2})\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2{\it\alpha}(\mathbf{x}_{1}-\mathbf{x}_{2})^{T}B\mathbf{y})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant X}}\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X \\ 1\leqslant \mathbf{x}+\mathbf{h}\leqslant X}}{\it\xi}(\mathbf{x}+\mathbf{h},\mathbf{x})\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2{\it\alpha}(\mathbf{h}^{T}B\mathbf{y}))\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,X^{r}L^{2r}\mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2{\it\alpha}(\mathbf{h}^{T}B\mathbf{y}))\bigg|,\nonumber\end{eqnarray}$$
where
${\it\xi}(\mathbf{x}_{1},\mathbf{x}_{2})$
is defined as
$$\begin{eqnarray}{\it\xi}(\mathbf{x}_{1},\mathbf{x}_{2})={\rm\Lambda}(\mathbf{x}_{1}){\rm\Lambda}(\mathbf{x}_{2})e({\it\alpha}(\mathbf{x}_{1}^{T}A_{1}\mathbf{x}_{1}-\mathbf{x}_{2}^{T}A_{1}\mathbf{x}_{2})).\end{eqnarray}$$
We write
$$\begin{eqnarray}B=\left(\begin{array}{@{}ccc@{}}b_{1,1} & \cdots \, & b_{1,s}\\ \vdots & \cdots \, & \vdots \\ b_{r,1} & \cdots \, & b_{r,s}\end{array}\right).\end{eqnarray}$$
Since
$\text{rank}(B)\geqslant 3$
, without loss of generality, we assume that
$\text{rank}(B_{0})=3$
, where
$B_{0}=(b_{i,j})_{1\leqslant i,j\leqslant 3}$
. Let
$B^{\prime }=(b_{i,j})_{4\leqslant i\leqslant r,1\leqslant j\leqslant 3}$
. Then one has
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2\mathbf{h}^{T}B\mathbf{y}{\it\alpha})\bigg|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,X^{s-3}\mathop{\sum }_{|h_{4}|,\ldots ,|h_{r}|\leqslant X}\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{v}\leqslant X}e(2{\it\alpha}(\mathbf{u}^{T}B_{0}+\mathbf{k}^{T})\mathbf{v})\bigg|,\nonumber\end{eqnarray}$$
where
$\mathbf{u}^{T}=(h_{1},h_{2},h_{3})$
,
$\mathbf{v}^{T}=(y_{1},y_{2},y_{3})$
and
$\mathbf{k}^{T}=(h_{4},\ldots ,h_{r})B^{\prime }$
. By changing variables
$\mathbf{x}^{T}=2(\mathbf{u}^{T}B_{0}+\mathbf{k}^{T})$
, we obtain
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2\mathbf{h}^{T}B\mathbf{y}{\it\alpha})\bigg| & {\leqslant} & \displaystyle X^{s-3}\mathop{\sum }_{|h_{4}|,\ldots ,|h_{r}|\leqslant X}\mathop{\sum }_{\substack{ |\mathbf{x}|\ll X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{v}\leqslant X}e({\it\alpha}(\mathbf{x}^{T}\mathbf{v}))\bigg|\nonumber\\ \displaystyle & \ll & \displaystyle X^{r+s-6}\mathop{\sum }_{\substack{ |\mathbf{x}|\ll X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{v}\leqslant X}e({\it\alpha}(\mathbf{x}^{T}\mathbf{v}))\bigg|.\nonumber\end{eqnarray}$$
We apply Lemma 4.2 to conclude that
$$\begin{eqnarray}\mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e(2\mathbf{h}^{T}B\mathbf{y}{\it\alpha})\bigg|\ll X^{r+s}Q^{-3}L^{3},\end{eqnarray}$$
and therefore,
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{x}\leqslant X}}{\rm\Lambda}(\mathbf{x})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}))\bigg|^{2}\ll X^{2r+s}Q^{-3}L^{2r+3}.\end{eqnarray}$$
Similar to (4.7), we can prove
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}\bigg|\mathop{\sum }_{\substack{ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{z})e({\it\alpha}(2\mathbf{y}^{T}C\mathbf{z}+\mathbf{z}^{T}A_{3}\mathbf{z}))\bigg|^{2}\ll X^{2t+s}Q^{-2}L^{2t+2}.\end{eqnarray}$$
Lemma 4.6. Suppose that
$A$
is in the form (4.4) with
$\text{rank}(B)\geqslant 3$
and
$\text{rank}(C)\geqslant 2$
. Then we have
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/3}.\end{eqnarray}$$
Proof. By Dirichlet’s approximation theorem, for any
${\it\alpha}\in [X^{-1},1+X^{-1}]$
, there exist
$a$
and
$q$
with
$1\leqslant a\leqslant q\leqslant X$
and
$(a,q)=1$
such that
$|{\it\alpha}-a/q|\leqslant (qX)^{-1}$
. Thus the desired conclusion follows from Lemma 4.4 by the dyadic argument.◻
5 Quadratic forms with off-diagonal rank
${\leqslant}3$
Proposition 5.1. Let
$A$
be given by (1.1), and let
$S({\it\alpha})$
be defined in (2.5). Suppose that
$\text{rank}(A)\geqslant 9$
and
$\text{rank}_{\text{off}}(A)\leqslant 3$
. Then we have
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,\,d{\it\alpha}\ll X^{n-2}L^{-K/6},\end{eqnarray}$$
where the implied constant depends on
$A$
and
$K$
.
From now on, we assume throughout Section 5 that
$\text{rank}(A)\geqslant 9$
and
$$\begin{eqnarray}\text{rank}_{\text{off}}(A)=\text{rank}(B)=3,\end{eqnarray}$$
where
$$\begin{eqnarray}B=\left(\begin{array}{@{}ccc@{}}a_{1,4} & a_{1,5} & a_{1,6}\\ a_{2,4} & a_{2,5} & a_{2,6}\\ a_{3,4} & a_{3,5} & a_{3,6}\end{array}\right).\end{eqnarray}$$
Then we introduce
$B_{1},B_{2},B_{3}\in M_{3,n-4}(\mathbb{Z})$
defined as
$$\begin{eqnarray}\displaystyle B_{1} & = & \displaystyle \left(\begin{array}{@{}cccccc@{}}a_{1,5} & a_{1,6} & a_{1,7} & a_{1,8} & \cdots \, & a_{1,n}\\ a_{2,5} & a_{2,6} & a_{2,7} & a_{2,8} & \cdots \, & a_{2,n}\\ a_{3,5} & a_{3,6} & a_{3,7} & a_{3,8} & \cdots \, & a_{3,n}\end{array}\right),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle B_{2} & = & \displaystyle \left(\begin{array}{@{}cccccc@{}}a_{1,4} & a_{1,6} & a_{1,7} & a_{1,8} & \cdots \, & a_{1,n}\\ a_{2,4} & a_{2,6} & a_{2,7} & a_{2,8} & \cdots \, & a_{2,n}\\ a_{3,4} & a_{3,6} & a_{3,7} & a_{3,8} & \cdots \, & a_{3,n}\end{array}\right),\end{eqnarray}$$
and
$$\begin{eqnarray}B_{3}=\left(\begin{array}{@{}cccccc@{}}a_{1,4} & a_{1,5} & a_{1,7} & a_{1,8} & \cdots \, & a_{1,n}\\ a_{2,4} & a_{2,5} & a_{2,7} & a_{2,8} & \cdots \, & a_{2,n}\\ a_{3,4} & a_{3,5} & a_{3,7} & a_{3,8} & \cdots \, & a_{3,n}\end{array}\right).\end{eqnarray}$$
Subject to the assumption (5.1), we have the following.
Lemma 5.2. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=\text{rank}(B_{3})=2$
, then one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Lemma 5.3. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=2$
and
$\text{rank}(B_{3})=3$
, then one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Lemma 5.4. If
$\text{rank}(B_{1})=2$
and
$\text{rank}(B_{2})=\text{rank}(B_{3})=3$
, then one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Lemma 5.5. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=\text{rank}(B_{3})=3$
, then one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Remark for the Proof of Proposition 5.1.
If
$\text{rank}_{\text{off}}(A)=0$
, then
$A$
is a diagonal matrix and the conclusion is classical. When
$\text{rank}_{\text{off}}(A)=3$
, our conclusion follows from Lemmas 5.2–5.5 immediately. The method applied to establish Lemmas 5.2–5.5 can also be used to deal with the case
$1\leqslant \text{rank}_{\text{off}}(A)\leqslant 2$
. Indeed, the proof of Proposition 5.1 under the condition
$1\leqslant \text{rank}_{\text{off}}(A)\leqslant 2$
is easier, and we omit the details. Therefore, our main task is to establish Lemmas 5.2–5.5.
Lemma 5.6. Let
$C\in M_{n,n}(\mathbb{Q})$
be a symmetric matrix, and let
$H\in M_{n,k}(\mathbb{Q})$
. For
${\it\alpha}\in \mathbb{R}$
and
${\bf\beta}\in \mathbb{R}^{k}$
, we define
$$\begin{eqnarray}{\mathcal{F}}({\it\alpha},\,{\bf\beta})=\mathop{\sum }_{\mathbf{x}\in \,{\mathcal{X}}}w(\mathbf{x})e({\it\alpha}\mathbf{x}^{T}C\mathbf{x}+\mathbf{x}^{T}H{\bf\beta}),\end{eqnarray}$$
where
${\mathcal{X}}\subset \mathbb{Z}^{n}$
is a finite subset of
$\mathbb{Z}^{n}$
. Let
$$\begin{eqnarray}{\mathcal{N}}({\mathcal{F}})=\mathop{\sum }_{\substack{ \mathbf{x}\in \,{\mathcal{X}},\,\mathbf{y}\in \,{\mathcal{X}} \\ \mathbf{x}^{T}C\mathbf{x}=\mathbf{y}^{T}C\mathbf{y} \\ \mathbf{x}^{T}H=\mathbf{y}^{T}H}}w(\mathbf{x})w(\mathbf{y}).\end{eqnarray}$$
Then we have
$$\begin{eqnarray}\int _{[0,1]^{k+1}}|{\mathcal{F}}({\it\alpha},\,{\bf\beta})|^{2}\,\,d{\it\alpha}\,\,d{\bf\beta}\ll {\mathcal{N}}({\mathcal{F}}),\end{eqnarray}$$
where the implied constant may depend on
$C$
and
$H$
.
Proof. We can choose a natural number
$h\in \mathbb{N}$
such that
$hC\in M_{n,n}(\mathbb{Z})$
and
$hH\in M_{n,k}(\mathbb{Z})$
. Then we deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[0,1]^{k+1}}|{\mathcal{F}}({\it\alpha},\,{\bf\beta})|^{2}\,d{\it\alpha}\,d{\bf\beta}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \int _{[0,h]^{k+1}}\bigg|\mathop{\sum }_{\mathbf{x}\in \,{\mathcal{X}}}w(\mathbf{x})e(h^{-1}{\it\alpha}\mathbf{x}^{T}(hC)\mathbf{x}+\mathbf{x}^{T}(hH)(h^{-1}{\bf\beta}))\bigg|^{2}\,d{\it\alpha}\,d{\bf\beta}\nonumber\\ \displaystyle & & \displaystyle \quad =h^{k+1}\int _{[0,1]^{k+1}}\bigg|\mathop{\sum }_{\mathbf{x}\in \,{\mathcal{X}}}w(\mathbf{x})e({\it\alpha}\mathbf{x}^{T}(hC)\mathbf{x}+\mathbf{x}^{T}(hH){\bf\beta})\bigg|^{2}\,d{\it\alpha}\,d{\bf\beta}.\nonumber\end{eqnarray}$$
By orthogonality, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[0,1]^{k+1}}\bigg|\mathop{\sum }_{\mathbf{x}\in \,{\mathcal{X}}}w(\mathbf{x})e({\it\alpha}\mathbf{x}^{T}(hC)\mathbf{x}+\mathbf{x}^{T}(hH){\bf\beta})\bigg|^{2}\,d{\it\alpha}\,d{\bf\beta}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ \mathbf{x}\in \,{\mathcal{X}},~\mathbf{y}\in \,{\mathcal{X}} \\ \mathbf{x}^{T}(hC)\mathbf{x}=\mathbf{y}^{T}(hC)\mathbf{y} \\ \mathbf{x}^{T}(hH)=\mathbf{y}^{T}(hH)}}w(\mathbf{x})w(\mathbf{y})={\mathcal{N}}({\mathcal{F}}).\nonumber\end{eqnarray}$$
Therefore, one obtains
$$\begin{eqnarray}\int _{[0,1]^{k+1}}|{\mathcal{F}}({\it\alpha},\,{\bf\beta})|^{2}\,d{\it\alpha}\,d{\bf\beta}\leqslant h^{k+1}{\mathcal{N}}({\mathcal{F}}),\end{eqnarray}$$
and this completes the proof. ◻
Lemma 5.7. Let
$C\in M_{n,n}(\mathbb{Q})$
be a symmetric matrix, and let
$H\in M_{n,k}(\mathbb{Q})$
. We have
$$\begin{eqnarray}{\mathcal{N}}_{1}\ll {\mathcal{N}}_{2},\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{N}}_{1}=\mathop{\sum }_{\substack{ |\mathbf{x}|\ll X,\,|\mathbf{y}|\ll X \\ \mathbf{x}^{T}C\mathbf{x}=\mathbf{y}^{T}C\mathbf{y} \\ \mathbf{x}^{T}H=\mathbf{y}^{T}H}}1\quad \text{and}\quad {\mathcal{N}}_{2}=\mathop{\sum }_{\substack{ |\mathbf{x}|\ll X,\,|\mathbf{y}|\ll X \\ \mathbf{x}^{T}C\mathbf{y}=0 \\ \mathbf{x}^{T}H=0}}1.\end{eqnarray}$$
Proof. By changing variables
$\mathbf{x}-\mathbf{y}=\mathbf{h}$
and
$\mathbf{x}+\mathbf{y}=\mathbf{z}$
, the desired conclusion follows immediately.◻
The following result is well known.
Lemma 5.8. Let
$C\in M_{k,m}(\mathbb{Q})$
. If
$\text{rank}(C)\geqslant 2$
, then one has
$$\begin{eqnarray}\mathop{\sum }_{\substack{ |\mathbf{x}|\ll X,|\mathbf{y}|\ll X \\ \mathbf{x}^{T}C\mathbf{y}=0}}1\ll X^{k+m-2}L,\end{eqnarray}$$
where the implied constant depends on the matrix
$C$
.
5.1 Proof of Lemma 5.2
Lemma 5.9. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=\text{rank}(B_{3})=2$
, then we can write
$A$
in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & B & 0\\ B^{T} & A_{2} & C\\ 0 & C^{T} & D\end{array}\right),\end{eqnarray}$$
where
$B\in GL_{3}(\mathbb{Z})$
,
$C\in M_{3,n-6}(\mathbb{Z})$
and
$D=\text{diag}\{d_{1},\ldots ,d_{n-6}\}$
is a diagonal matrix.
Proof. We write for
$1\leqslant j\leqslant n-3$
that
$$\begin{eqnarray}{\it\gamma}_{j}=\left(\begin{array}{@{}c@{}}a_{1,\,3+j}\\ a_{2,\,3+j}\\ a_{3,\,3+j}\end{array}\right).\end{eqnarray}$$
Since
$B=({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\in GL_{3}(\mathbb{Z})$
,
${\it\gamma}_{1}$
,
${\it\gamma}_{2}$
and
${\it\gamma}_{3}$
are linearly independent. For any
$4\leqslant j\leqslant n-3$
, one has
$\text{rank}({\it\gamma}_{2},{\it\gamma}_{3},{\it\gamma}_{j})\leqslant \text{rank}(B_{1})=2$
. Therefore, we obtain
${\it\gamma}_{j}\in <{\it\gamma}_{2},{\it\gamma}_{3}>$
. Similarly, one has
${\it\gamma}_{j}\in <{\it\gamma}_{1},{\it\gamma}_{3}>$
and
${\it\gamma}_{j}\in <{\it\gamma}_{1},{\it\gamma}_{2}>$
. Then we can conclude that
${\it\gamma}_{j}=0$
for
$4\leqslant j\leqslant n-3$
.
For
$7\leqslant i<j\leqslant n$
, we write
$$\begin{eqnarray}B_{i,j}=\left(\begin{array}{@{}cccc@{}}a_{1,4} & a_{1,5} & a_{1,6} & a_{1,j}\\ a_{2,4} & a_{2,5} & a_{2,6} & a_{2,j}\\ a_{3,4} & a_{3,5} & a_{3,6} & a_{3,j}\\ a_{i,4} & a_{i,5} & a_{i,6} & a_{i,j}\end{array}\right)=\left(\begin{array}{@{}c@{}}{\it\eta}_{1}^{T}\\ {\it\eta}_{2}^{T}\\ {\it\eta}_{3}^{T}\\ {\it\eta}_{4}^{T}\end{array}\right).\end{eqnarray}$$
Since
$3\leqslant \text{rank}(B_{i,j})\leqslant \text{rank}_{\text{off}}(A)=3$
, we conclude that
${\it\eta}_{4}^{T}$
can be linearly represented by
${\it\eta}_{1}^{T}$
,
${\it\eta}_{2}^{T}$
and
${\it\eta}_{3}^{T}$
. Then we obtain
$a_{i,j}=0$
due to
$a_{1,j}=a_{2,j}=a_{3,j}=0$
. Therefore, the matrix
$A$
is in the form (5.6). We complete the proof.◻
Proof of Lemma 5.2.
By Lemma 5.9, we have
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{y}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-6} \\ 1\leqslant \mathbf{z}\leqslant X}}\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B\mathbf{y}+\mathbf{y}^{T}A_{2}\mathbf{y}+2\mathbf{z}^{T}C^{T}\mathbf{y}+\mathbf{z}^{T}D\mathbf{z}))\nonumber\\ \displaystyle & & \displaystyle \times \,{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z}).\nonumber\end{eqnarray}$$
By orthogonality, we have
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \int _{[0,1]^{3}}\mathop{\sum }_{\substack{ \mathbf{w}\in \mathbb{Z}^{3} \\ |\mathbf{w}|\ll X}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{y}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-6} \\ 1\leqslant \mathbf{z}\leqslant X}}e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+\mathbf{w}^{T}\mathbf{y}+\mathbf{z}^{T}D\mathbf{z}))\nonumber\\ \displaystyle & & \displaystyle \quad \times \,e((2\mathbf{x}^{T}B+\mathbf{y}^{T}A_{2}+2\mathbf{z}^{T}C^{T}-\mathbf{w}^{T}){\bf\beta}){\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z})\,d{\bf\beta},\nonumber\end{eqnarray}$$
where
${\bf\beta}=({\it\beta}_{1},{\it\beta}_{2},{\it\beta}_{3})^{T}$
and we use
$d{\bf\beta}$
to denote
$d{\it\beta}_{1}\,d{\it\beta}_{2}\,d{\it\beta}_{3}$
. We define
$$\begin{eqnarray}{\mathcal{F}}({\it\alpha},{\bf\beta})=\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}e({\it\alpha}\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}B{\bf\beta}){\rm\Lambda}(\mathbf{x}),\end{eqnarray}$$
and
$$\begin{eqnarray}f_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{\substack{ 1\leqslant z\leqslant X}}e({\it\alpha}d_{j}z^{2}+2z{\it\xi}_{j}^{T}{\bf\beta}){\rm\Lambda}(z),\end{eqnarray}$$
where
${\it\xi}_{j}=(a_{4,6+j},a_{5,6+j},a_{6,6+j})^{T}$
for
$1\leqslant j\leqslant n-6$
. On writing
$I_{3}=(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3})$
, we introduce
$$\begin{eqnarray}{\mathcal{H}}_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{|w|\ll X}\mathop{\sum }_{\substack{ 1\leqslant y\leqslant X}}e({\it\alpha}wy+y{\it\gamma}_{j}^{T}{\bf\beta}-w\mathbf{e}_{j}^{T}{\bf\beta}){\rm\Lambda}(y),\end{eqnarray}$$
where
${\it\gamma}_{j}^{T}=(a_{3+j,4},a_{3+j,5},a_{3+j,6})$
for
$1\leqslant j\leqslant 3$
. With above notations, we have
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}}S({\it\alpha})\,d{\it\alpha} & = & \displaystyle \int _{\mathfrak{m}}\int _{[0,1]^{3}}{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{j=1}^{n-6}f_{j}({\it\alpha},{\bf\beta})\,d{\bf\beta}\,d{\it\alpha}.\nonumber\end{eqnarray}$$
Therefore, one has the following inequality
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha} & {\leqslant} & \displaystyle \int _{\mathfrak{m}}\int _{[0,1]^{3}}\bigg|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{j=1}^{n-6}f_{j}({\it\alpha},{\bf\beta})\bigg|\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
We first consider the case
$\text{rank}(D)\geqslant 3$
. Without loss of generality, we assume
$d_{1}d_{2}d_{3}\not =0$
. By (5.8) and the Cauchy–Schwarz inequality, one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant \,{\mathcal{I}}_{1}^{1/2}{\mathcal{I}}_{2}^{1/2}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\Big|\mathop{\prod }_{j=3}^{n-6}f_{j}({\it\alpha},{\bf\beta})\bigg|,\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{I}}_{1}=\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})f_{1}({\it\alpha},{\bf\beta})f_{2}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{I}}_{2}=\int _{[0,1]^{4}}|{\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
By Lemmas 5.6 and 5.7, one has
$$\begin{eqnarray}\displaystyle {\mathcal{I}}_{1} & \ll & \displaystyle L^{10}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|z_{1}|,|z_{1}^{\prime }|,|z_{2}|,|z_{2}^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}+d_{1}z_{1}^{2}+d_{2}z_{2}^{2}=\mathbf{x}^{\prime T}A_{1}\mathbf{x}^{\prime }+d_{1}z_{1}^{\prime 2}+d_{2}z_{2}^{\prime 2} \\ \mathbf{x}^{T}B+z_{1}{\it\gamma}_{1}^{T}+z_{2}{\it\gamma}_{2}^{T}=\mathbf{x}^{\prime T}B+z_{1}^{\prime }{\it\gamma}_{1}^{T}+z_{2}^{\prime }{\it\gamma}_{2}^{T}}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{10}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|z_{1}|,|z_{1}^{\prime }|,|z_{2}|,|z_{2}^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+d_{1}z_{1}z_{1}^{\prime }+d_{2}z_{2}z_{2}^{\prime }=0 \\ \mathbf{x}^{T}B+z_{1}{\it\gamma}_{1}^{T}+z_{2}{\it\gamma}_{2}^{T}=0}}1.\nonumber\end{eqnarray}$$
Since
$B$
is invertible, we obtain
$$\begin{eqnarray}{\mathcal{I}}_{1}\ll L^{10}\mathop{\sum }_{\substack{ |\mathbf{x}^{\prime }|,|z_{1}|,|z_{1}^{\prime }|,|z_{2}|,|z_{2}^{\prime }|\ll X \\ -(z_{1}{\it\gamma}_{1}^{T}+z_{2}{\it\gamma}_{2}^{T})B^{-1}A_{1}\mathbf{x}^{\prime }+d_{1}z_{1}z_{1}^{\prime }+d_{2}z_{2}z_{2}^{\prime }=0}}1.\end{eqnarray}$$
Then we conclude from Lemma 5.8 that
$$\begin{eqnarray}{\mathcal{I}}_{1}\ll X^{5}L^{11}.\end{eqnarray}$$
It follows from Lemmas 5.6–5.7 that
$$\begin{eqnarray}\displaystyle {\mathcal{I}}_{2} & \ll & \displaystyle L^{6}\!\mathop{\sum }_{\substack{ |w_{1}|,|w_{1}^{\prime }|,|w_{2}|,|w_{2}^{\prime }|,|w_{3}|,|w_{3}^{\prime }|,|y_{1}|,|y_{1}^{\prime }|,|y_{2}|,|y_{2}^{\prime }|,|y_{3}|,|y_{3}^{\prime }|\ll X \\ w_{1}y_{1}+w_{2}y_{2}+w_{3}y_{3}=w_{1}^{\prime }y_{1}^{\prime }+w_{2}^{\prime }y_{2}^{\prime }+w_{3}^{\prime }y_{3}^{\prime } \\ y_{1}{\it\gamma}_{1}^{T}-w_{1}\mathbf{e}_{1}^{T}+y_{2}{\it\gamma}_{2}^{T}-w_{2}\mathbf{e}_{2}^{T}+y_{3}{\it\gamma}_{3}^{T}-w_{3}\mathbf{e}_{3}^{T}=y_{1}^{\prime }{\it\gamma}_{1}^{T}-w_{1}^{\prime }\mathbf{e}_{1}^{T}+y_{2}^{\prime }{\it\gamma}_{2}^{T}-w_{2}^{\prime }\mathbf{e}_{2}^{T}+y_{3}^{\prime }{\it\gamma}_{3}^{T}-w_{3}^{\prime }\mathbf{e}_{3}^{T}}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{6}\mathop{\sum }_{\substack{ |w_{1}|,|w_{1}^{\prime }|,|w_{2}|,|w_{2}^{\prime }|,|w_{3}|,|w_{3}^{\prime }|,|y_{1}|,|y_{1}^{\prime }|,|y_{2}|,|y_{2}^{\prime }|,|y_{3}|,|y_{3}^{\prime }|\ll X \\ w_{1}y_{1}^{\prime }+w_{1}^{\prime }y_{1}+w_{2}y_{2}^{\prime }+w_{2}^{\prime }y_{2}+w_{3}y_{3}^{\prime }+w_{3}^{\prime }y_{3}=0 \\ y_{1}{\it\gamma}_{1}^{T}-w_{1}\mathbf{e}_{1}^{T}+y_{2}{\it\gamma}_{2}^{T}-w_{2}\mathbf{e}_{2}^{T}+y_{3}{\it\gamma}_{3}^{T}-w_{3}\mathbf{e}_{3}^{T}=0}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{6}\mathop{\sum }_{\substack{ |w_{1}^{\prime }|,|w_{2}^{\prime }|,|w_{3}^{\prime }|,|y_{1}|,|y_{1}^{\prime }|,|y_{2}|,|y_{2}^{\prime }|,|y_{3}|,|y_{3}^{\prime }|\ll X \\ \mathbf{y}^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})^{T}\mathbf{y}^{\prime }+\mathbf{y}^{T}\mathbf{w}^{\prime }=\mathbf{0}}}1,\nonumber\end{eqnarray}$$
where
$\mathbf{y}=(y_{1},y_{2},y_{3})^{T}$
,
$\mathbf{y}^{\prime }=(y_{1}^{\prime },y_{2}^{\prime },y_{3}^{\prime })^{T}$
and
$\mathbf{w}^{\prime }=(w_{1}^{\prime },w_{2}^{\prime },w_{3}^{\prime })^{T}$
. Then by Lemma 5.8, we have
$$\begin{eqnarray}{\mathcal{I}}_{2}\ll X^{7}L^{7}.\end{eqnarray}$$
Since
$d_{3}\not =0$
, we obtain by Lemma 4.3
$$\begin{eqnarray}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}|f_{3}({\it\alpha},{\bf\beta})|\ll XL^{-K/5},\end{eqnarray}$$
and thereby
$$\begin{eqnarray}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{j=3}^{n-6}f_{j}({\it\alpha},{\bf\beta})\bigg|\ll X^{n-8}L^{-K/5}.\end{eqnarray}$$
Now we conclude from (5.9), (5.12)–(5.14) that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Next we consider the case
$1\leqslant \text{rank}(D)\leqslant 2$
. Without loss of generality, we suppose that
$d_{1}\not =0$
and
$d_{k}=0$
for
$3\leqslant k\leqslant n$
. Since
$\text{rank}(A)\geqslant 9$
, there exists
$k$
with
$3\leqslant k\leqslant n-6$
such that
${\it\xi}_{k}\not =0\in \mathbb{Z}^{3}$
. Then we can find
$i,j$
with
$1\leqslant i<j\leqslant 3$
so that
$\text{rank}(\mathbf{e}_{i},\mathbf{e}_{j},{\it\xi}_{k})=3$
. Without loss of generality, we can assume that
$i=1,j=2$
and
$k=3$
. One has
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha} & {\leqslant} & \displaystyle \sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{j\not =3}f_{j}({\it\alpha},{\bf\beta})\bigg|\bigg(\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg(\int _{[0,1]^{4}}|{\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta})f_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}.\nonumber\end{eqnarray}$$
We deduce from Lemmas 5.6–5.7 that
$$\begin{eqnarray}\displaystyle \int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha} & \ll & \displaystyle L^{8}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|w|,|w^{\prime }|,|y|,|y^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}+wy={\mathbf{x}^{\prime }}^{T}A_{1}\mathbf{x}^{\prime }+w^{\prime }y^{\prime } \\ 2\mathbf{x}^{T}B+y{\it\gamma}_{3}^{T}-w\mathbf{e}_{3}^{T}=2{\mathbf{x}^{\prime }}^{T}B+y^{\prime }{\it\gamma}_{3}^{T}-w^{\prime }\mathbf{e}_{3}^{T}}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{8}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|w|,|w^{\prime }|,|y|,|y^{\prime }|\ll X \\ 2\mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+wy^{\prime }+w^{\prime }y=0 \\ 2\mathbf{x}^{T}B+y{\it\gamma}_{3}^{T}-w\mathbf{e}_{3}^{T}=0}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{8}\mathop{\sum }_{\substack{ |\mathbf{x}^{\prime }|,|w|,|w^{\prime }|,|y|,|y^{\prime }|\ll X \\ -(y{\it\gamma}_{3}^{T}-w\mathbf{e}_{3}^{T})B^{-1}A_{1}\mathbf{x}^{\prime }+wy^{\prime }+w^{\prime }y=0}}1.\nonumber\end{eqnarray}$$
Then by Lemma 5.8, one has
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{9}.\end{eqnarray}$$
We deduce from Lemmas 5.6–5.7 again that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[0,1]^{4}}|{\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta})f_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{6}\mathop{\sum }_{\substack{ |w_{1}|,|w_{1}^{\prime }|,|w_{2}|,|w_{2}^{\prime }|,|y_{1}|,|y_{1}^{\prime }|,|y_{2}|,|y_{2}^{\prime }|,|z|,|z^{\prime }|\ll X \\ w_{1}y_{1}+w_{2}y_{2}+d_{3}z^{2}=w_{1}^{\prime }y_{1}^{\prime }+w_{2}^{\prime }y_{2}^{\prime }+d_{3}{z^{\prime }}^{2} \\ y_{1}{\it\gamma}_{1}^{T}+y_{2}{\it\gamma}_{2}^{T}-w_{1}\mathbf{e}_{1}^{T}-w_{2}\mathbf{e}_{2}^{T}+2z{\it\xi}_{3}^{T}=y_{1}^{\prime }{\it\gamma}_{1}^{T}+y_{2}^{\prime }{\it\gamma}_{2}^{T}-w_{1}^{\prime }\mathbf{e}_{1}^{T}-w_{2}^{\prime }\mathbf{e}_{2}^{T}+2z^{\prime }{\it\xi}_{3}^{T}}}1\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{6}\mathop{\sum }_{\substack{ |w_{1}|,|w_{1}^{\prime }|,|w_{2}|,|w_{2}^{\prime }|,|y_{1}|,|y_{1}^{\prime }|,|y_{2}|,|y_{2}^{\prime }|,|z|,|z^{\prime }|\ll X \\ w_{1}y_{1}^{\prime }+w_{1}^{\prime }y_{1}+w_{2}y_{2}^{\prime }+w_{2}^{\prime }y_{2}+2d_{3}zz^{\prime }=0 \\ y_{1}{\it\gamma}_{1}^{T}+y_{2}{\it\gamma}_{2}^{T}-w_{1}\mathbf{e}_{1}^{T}-w_{2}\mathbf{e}_{2}^{T}+2z{\it\xi}_{3}^{T}=0}}1.\nonumber\end{eqnarray}$$
On applying
$\text{rank}(\mathbf{e}_{1},\mathbf{e}_{2},{\it\xi}_{3})=3$
and Lemma 5.8, we obtain
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta})f_{3}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{7}.\end{eqnarray}$$
It follows from Lemma 4.3 that
$$\begin{eqnarray}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{j\not =3}f_{j}({\it\alpha},{\bf\beta})\bigg|\ll X^{n-7}L^{-K/5}.\end{eqnarray}$$
Then we conclude from (5.15)–(5.17) that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Now it suffices to assume
$D=0$
. Then the matrix
$A$
is in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & B & 0\\ B^{T} & A_{2} & C\\ 0 & C^{T} & 0\end{array}\right).\end{eqnarray}$$
It follows from
$\text{rank}(A)\geqslant 9$
that
$\text{rank}(C)\geqslant 3$
. By Lemma 4.6,
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/3}.\end{eqnarray}$$
This completes the proof of Lemma 5.2.
5.2 Proof of Lemma 5.3
Lemma 5.10. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=2$
and
$\text{rank}(B_{3})=3$
, then the symmetric integral matrix
$A$
can be written in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & C & {\it\gamma}_{3}{\it\xi}^{T}\\ C^{T} & A_{2} & V\\ {\it\xi}{\it\gamma}_{3}^{T} & V^{T} & D+h{\it\xi}{\it\xi}^{T}\end{array}\right),\end{eqnarray}$$
where
$C=({\it\gamma}_{1},{\it\gamma}_{2})\in M_{3,2}(\mathbb{Z})$
,
${\it\gamma}_{3}\in \mathbb{Q}^{3}$
,
${\it\xi}\in \mathbb{Z}^{n-5}$
,
$V\in M_{2,n-5}(\mathbb{Z})$
,
$h\in \mathbb{Q}$
and
$D=\text{diag}\{d_{1},\ldots ,d_{n-5}\}\in M_{n-5,n-5}(\mathbb{Q})$
is a diagonal matrix. Moreover, one has
$({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\in GL_{3}(\mathbb{Q})$
.
Proof. Let us write
$$\begin{eqnarray}{\it\gamma}_{j}^{\prime }=\left(\begin{array}{@{}c@{}}a_{1,\,3+j}\\ a_{2,\,3+j}\\ a_{3,\,3+j}\end{array}\right)\quad \text{for }1\leqslant j\leqslant n-3.\end{eqnarray}$$
Since
$\text{rank}({\it\gamma}_{1}^{\prime },{\it\gamma}_{2}^{\prime },{\it\gamma}_{3}^{\prime })=\text{rank}(B)=3$
, we conclude that
${\it\gamma}_{1}^{\prime }$
,
${\it\gamma}_{2}^{\prime }$
and
${\it\gamma}_{3}^{\prime }$
are linearly independent. For any
$4\leqslant j\leqslant n-3$
, we deduce from
$\text{rank}(B_{1})=\text{rank}(B_{2})=2$
that
${\it\gamma}_{j}^{\prime }\in <{\it\gamma}_{2}^{\prime },{\it\gamma}_{3}^{\prime }>\cap <{\it\gamma}_{1}^{\prime },{\it\gamma}_{3}^{\prime }>=<{\it\gamma}_{3}^{\prime }>$
. Therefore, we can write
$A$
in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & C & {\it\gamma}_{3}{\it\xi}^{T}\\ C^{T} & A_{2} & V\\ {\it\xi}{\it\gamma}_{3}^{T} & V^{T} & A_{3}\end{array}\right),\end{eqnarray}$$
where
$C=({\it\gamma}_{1},{\it\gamma}_{2})\in M_{3,2}(\mathbb{Z})$
,
${\it\gamma}_{3}\in \mathbb{Q}^{3}$
,
${\it\xi}\in \mathbb{Z}^{n-5}$
,
$V\in M_{2,n-5}(\mathbb{Z})$
and
$A_{3}\in M_{n-5,n-5}(\mathbb{Q})$
.
For
$6\leqslant j\leqslant n$
. we define
${\it\eta}_{j}^{T}=(a_{j,4},\ldots ,a_{j,j-1},a_{j,j+1},\ldots ,a_{j,n})^{T}\in \mathbb{Z}^{n-4}$
. Then we set
${\it\theta}_{i,j}^{T}=(a_{i,4},\ldots ,a_{i,j-1},a_{i,j+1},\ldots ,a_{i,n})^{T}\in \mathbb{Z}^{n-4}$
for
$1\leqslant i\leqslant 3$
. Since
$\text{rank}_{\text{off}}(A)=\text{rank}(B)=\text{rank}(B_{3})=3$
,
${\it\eta}_{j}$
can be linearly represented by
${\it\theta}_{1,j}$
,
${\it\theta}_{2,j}$
and
${\it\theta}_{3,j}$
. Let
$$\begin{eqnarray}{\it\theta}_{i}^{T}=(a_{i,4},\ldots ,a_{i,n})^{T}\in \mathbb{Z}^{n-3}\quad \text{for }1\,\leqslant \,i\,\leqslant \,3.\end{eqnarray}$$
Then one can choose
$a_{j,j}^{\prime }\in \mathbb{Q}$
such that
$(a_{j,4},\ldots ,a_{j,j-1},a_{j,j}^{\prime },a_{j,j+1},\ldots ,a_{j,n})$
is linearly represented by
${\it\theta}_{1}$
,
${\it\theta}_{2}$
and
${\it\theta}_{3}$
. We consider
$A_{3}$
and
$A_{3}^{\prime }$
defined as
$$\begin{eqnarray}A_{3}=\left(\begin{array}{@{}ccc@{}}a_{6,6} & \cdots \, & a_{6,n}\\ \vdots & \cdots \, & \vdots \\ a_{n,6} & \cdots \, & a_{n,n}\end{array}\right)\quad \text{and}\quad A_{3}^{\prime }=\left(\begin{array}{@{}ccc@{}}a_{6,6}^{\prime } & \cdots \, & a_{6,n}^{\prime }\\ \vdots & \cdots \, & \vdots \\ a_{n,6}^{\prime } & \cdots \, & a_{n,n}^{\prime }\end{array}\right),\end{eqnarray}$$
where
$a_{i,j}^{\prime }=a_{i,j}$
for
$6\leqslant i\not =j\leqslant n$
. Since
$A_{3}^{\prime }$
is symmetric, we conclude from above that
$A_{3}^{\prime }=h{\it\xi}{\it\xi}^{T}$
for some
$h\in \mathbb{Q}$
. The proof is completed by noting that
$D=A_{3}-A_{3}^{\prime }$
is a diagonal matrix.◻
Proof of Lemma 5.3.
One can deduce from Lemma 5.10 that
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{2} \\ 1\leqslant \mathbf{y}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-5} \\ 1\leqslant \mathbf{z}\leqslant X}}e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}{\it\gamma}_{3}{\it\xi}^{T}\mathbf{z}+\mathbf{z}^{T}D\mathbf{z}+h\mathbf{z}^{T}{\it\xi}{\it\xi}^{T}\mathbf{z}))\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(2\mathbf{x}^{T}C\mathbf{y}+\mathbf{y}^{T}A_{2}\mathbf{y}+2\mathbf{z}^{T}V^{T}\mathbf{y})){\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z}).\nonumber\end{eqnarray}$$
We introduce new variables
$\mathbf{w}\in \mathbb{Z}^{2}$
and
$s\in \mathbb{Z}$
to replace
$2\mathbf{x}^{T}C+\mathbf{y}^{T}A_{2}+2\mathbf{z}^{T}V^{T}$
and
${\it\xi}^{T}\mathbf{z}$
, respectively. Therefore, we have
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \int _{[0,1]^{3}}\mathop{\sum }_{|s|\ll X}\mathop{\sum }_{\substack{ \mathbf{w}\in \mathbb{Z}^{2} \\ |\mathbf{w}|\ll X}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{2} \\ 1\leqslant \mathbf{y}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-5} \\ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z})\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+\mathbf{w}^{T}\mathbf{y}+\mathbf{z}^{T}D\mathbf{z}+2\mathbf{x}^{T}{\it\gamma}_{3}s+hs^{2}))\nonumber\\ \displaystyle & & \displaystyle \times \,e((2\mathbf{x}^{T}C+\mathbf{y}^{T}A_{2}+2\mathbf{z}^{T}V^{T}-\mathbf{w}^{T}){\bf\beta}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \times \,e(({\it\xi}^{T}\mathbf{z}-s){\it\beta}_{3})\,d{\bf\beta},\nonumber\end{eqnarray}$$
where
${\bf\beta}^{\prime }=({\it\beta}_{1},{\it\beta}_{2})^{T}$
,
${\bf\beta}=({\it\beta}_{1},{\it\beta}_{2},{\it\beta}_{3})^{T}$
and
$d{\bf\beta}=d{\it\beta}_{1}\,d{\it\beta}_{2}\,d{\it\beta}_{3}$
. We define
$$\begin{eqnarray}{\mathcal{F}}({\it\alpha},{\bf\beta})=\mathop{\sum }_{|s|\ll X}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}{\it\gamma}_{3}s+hs^{2})+2\mathbf{x}^{T}C{\bf\beta}^{\prime }-s{\it\beta}_{3}){\rm\Lambda}(\mathbf{x}).\end{eqnarray}$$
On writing
$I_{2}=(\mathbf{e}_{1},\mathbf{e}_{2})$
, we introduce
$$\begin{eqnarray}{\mathcal{H}}_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{|w|\ll X}\mathop{\sum }_{\substack{ 1\leqslant y\leqslant X}}e({\it\alpha}wy+y{\it\rho}_{j}^{T}{\bf\beta}^{\prime }-w\mathbf{e}_{j}^{T}{\bf\beta}^{\prime }){\rm\Lambda}(y),\end{eqnarray}$$
where
${\it\rho}_{j}=(a_{3+j,4},a_{3+j,5})^{T}$
for
$1\leqslant j\leqslant 2$
. Let
${\it\xi}=({\it\epsilon}_{1},\ldots ,{\it\epsilon}_{n-5})^{T}$
. Then we define
$$\begin{eqnarray}f_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{\substack{ 1\leqslant z\leqslant X}}e({\it\alpha}d_{j}z^{2}+2z{\it\upsilon}_{j}^{T}{\bf\beta}^{\prime }+{\it\epsilon}_{j}z{\it\beta}_{3}){\rm\Lambda}(z),\end{eqnarray}$$
where
$V=({\it\upsilon}_{1},\ldots ,{\it\upsilon}_{n-5})$
with
${\it\upsilon}_{j}=(a_{4,5+j},a_{5,5+j})^{T}$
for
$1\leqslant j\leqslant n-5$
. With above notations, we obtain
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant \int _{\mathfrak{m}}\int _{[0,1]^{3}}\bigg|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta})\mathop{\prod }_{j=1}^{n-5}f_{j}({\it\alpha},{\bf\beta})\bigg|\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
Let
$$\begin{eqnarray}{\mathcal{J}}_{1}=\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{J}}_{2}=\int _{[0,1]^{4}}|{\mathcal{H}}_{1}({\it\alpha},{\bf\beta}){\mathcal{H}}_{2}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
By (5.19) and the Cauchy–Schwarz inequality, one has for
$i\not =j$
that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant {\mathcal{J}}_{1}^{1/2}{\mathcal{J}}_{2}^{1/2}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{k\not =i,j}f_{k}({\it\alpha},{\bf\beta})\bigg|.\end{eqnarray}$$
One can deduce by Lemmas 5.6 and 5.7 that
$$\begin{eqnarray}{\mathcal{J}}_{1}\ll L^{8}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|s|,|s^{\prime }|,|z|,|z^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{x}^{T}{\it\gamma}_{3}s^{\prime }+s{\it\gamma}_{3}^{T}\mathbf{x}^{\prime }+hss^{\prime }+d_{i}zz^{\prime }=0 \\ \mathbf{x}^{T}C+z{\it\upsilon}_{i}^{T}=0 \\ s={\it\epsilon}_{i}z}}1.\end{eqnarray}$$
Note that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|s|,|s^{\prime }|,|z|,|z^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{x}^{T}{\it\gamma}_{3}s^{\prime }+s{\it\gamma}_{3}^{T}\mathbf{x}^{\prime }+hss^{\prime }+d_{i}zz^{\prime }=0 \\ \mathbf{x}^{T}C+z{\it\upsilon}_{i}^{T}=0 \\ s={\it\epsilon}_{i}z}}1 & = & \displaystyle \mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|s^{\prime }|,|z|,|z^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{x}^{T}{\it\gamma}_{3}s^{\prime }+{\it\epsilon}_{i}z{\it\gamma}_{3}^{T}\mathbf{x}^{\prime }+h{\it\epsilon}_{i}zs^{\prime }+d_{i}zz^{\prime }=0 \\ \mathbf{x}^{T}C+z{\it\upsilon}_{i}^{T}=0}}1\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|s|,|s^{\prime }|,|z|,|z^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+ss^{\prime }+{\it\epsilon}_{i}z{\it\gamma}_{3}^{T}\mathbf{x}^{\prime }+h{\it\epsilon}_{i}zs^{\prime }+d_{i}zz^{\prime }=0 \\ \mathbf{x}^{T}(C,{\it\gamma}_{3})+(z{\it\upsilon}_{i}^{T},-s)=0}}1.\nonumber\end{eqnarray}$$
Recalling
$\text{rank}(C,{\it\gamma}_{3})=3$
, one can replace
$\mathbf{x}$
by
$-(z{\it\upsilon}_{i}^{T},-s)(C,{\it\gamma}_{3})^{-1}$
. Therefore, by Lemma 5.8, one has
$$\begin{eqnarray}{\mathcal{J}}_{1}\ll X^{5}L^{9}\quad \text{if }d_{i}\not =0.\end{eqnarray}$$
The argument leading to (5.16) also implies
$$\begin{eqnarray}{\mathcal{J}}_{2}\ll X^{5}L^{5}\quad \text{if }{\it\epsilon}_{j}\not =0.\end{eqnarray}$$
Now we are able to handle the case
$\text{rank}(D)\geqslant 2$
. Since
$\text{rank}(B_{3})=3$
, one has
${\it\epsilon}_{l}\not =0$
for some
$l$
satisfying
$2\leqslant l\leqslant n-5$
. We may assume
${\it\epsilon}_{2}\not =0$
. We also have
${\it\epsilon}_{1}\not =0$
due to
$\text{rank}(B)=3$
. If
$d_{l}\not =0$
for some
$l\geqslant 3$
, then we can find
$i,j,k$
pairwise distinct so that
${\it\epsilon}_{j}\not =0$
and
$d_{i}d_{k}\not =0$
. If
$d_{1}d_{2}\not =0$
and
${\it\epsilon}_{j}\not =0$
for some
$j\geqslant 3$
, then we can also find
$i,j,k$
pairwise distinct so that
${\it\epsilon}_{j}\not =0$
and
$d_{i}d_{k}\not =0$
. In these cases, we can conclude from (5.20)–(5.22) that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
Next we assume
$d_{l}={\it\epsilon}_{l}=0$
for all
$l\geqslant 3$
. Then we can represent
$A$
in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & H & 0\\ H^{T} & Y & W\\ 0 & W^{T} & 0\end{array}\right),\end{eqnarray}$$
where
$H\in M_{3,4}(\mathbb{Z})$
,
$Y\in M_{4,4}(\mathbb{Z})$
and
$W\in M_{4,n-7}(\mathbb{Z})$
. It follows from
$\text{rank}(B)=3$
and
$\text{rank}(A)\geqslant 9$
that
$\text{rank}(H)\geqslant 3$
and
$\text{rank}(W)\geqslant 2$
. We apply Lemma 4.6 to conclude
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/3}.\end{eqnarray}$$
We are left to handle the case
$\text{rank}(D)\leqslant 1$
. Since
$\text{rank}(D)+\text{rank}(V)+1+5\geqslant \text{rank}(A)\geqslant 9$
, we obtain
$\text{rank}(D)\geqslant 1$
. Therefore,
$\text{rank}(D)=1$
. We have
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant {\mathcal{J}}_{3}^{1/2}{\mathcal{J}}_{4}^{1/2}\sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{u\not =i,j,k}f_{u}({\it\alpha},{\bf\beta})\bigg|,\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{J}}_{3}=\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})H_{1}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{J}}_{4}=\int _{[0,1]^{4}}|{\mathcal{H}}_{2}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})f_{k}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
$$\begin{eqnarray}{\mathcal{J}}_{3}\ll L^{8}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|s|,|s^{\prime }|,|y|,|y^{\prime }|,|w|,|w^{\prime }|\ll X \\ 2\mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+2\mathbf{x}^{T}{\it\gamma}_{3}s^{\prime }+2hss^{\prime }+2s{\it\gamma}_{3}^{T}\mathbf{x}^{\prime }+wy^{\prime }+yw^{\prime }=0 \\ 2\mathbf{x}^{T}C+y{\it\rho}_{1}^{T}-w\mathbf{e}_{1}^{T}=0 \\ s=0}}1.\end{eqnarray}$$
Since
$\text{rank}(C)\geqslant 3$
, we can represent two of
$x_{1},x_{2},x_{3}$
(say
$x_{1}$
and
$x_{2}$
) in terms of
$x_{3}$
,
$y$
and
$w$
. Then by Lemma 5.8, one has
$$\begin{eqnarray}{\mathcal{J}}_{3}\ll X^{7}L^{9}.\end{eqnarray}$$
We deduce from Lemma 5.10 that
$\text{rank}\left(\begin{array}{@{}c@{}}{\it\xi}^{T}\\ V\end{array}\right)\geqslant \text{rank}(A)-5-\text{rank}(D)\geqslant 3$
. Therefore, there exist distinct
$i,j,k,s$
such that
$\text{rank}\left(\begin{array}{@{}ccc@{}}{\it\upsilon}_{i}^{T} & {\it\upsilon}_{j}^{T} & {\it\upsilon}_{k}^{T}\\ {\it\epsilon}_{i} & {\it\epsilon}_{j} & {\it\epsilon}_{k}\end{array}\right)=3$
and
$d_{s}\not =0$
. By Lemmas 5.6–5.7, we also have
$$\begin{eqnarray}{\mathcal{J}}_{4}\ll L^{8}\mathop{\sum }_{\substack{ |y|,|y^{\prime }|,|w|,|w^{\prime }|,|z_{1}|,|z_{1}^{\prime }|,|z_{2}|,|z_{2}^{\prime }|,|z_{3}|,|z_{3}^{\prime }|\ll X \\ wy^{\prime }+yw^{\prime }+2d_{i}z_{1}z_{1}^{\prime }+2d_{j}z_{2}z_{2}^{\prime }+2d_{k}z_{3}z_{3}^{\prime }=0 \\ y{\it\rho}_{2}^{T}-w\mathbf{e}_{2}^{T}+2z_{1}{\it\upsilon}_{i}^{T}+2z_{2}{\it\upsilon}_{j}^{T}+2z_{3}{\it\upsilon}_{k}^{T}=0 \\ {\it\epsilon}_{i}z_{1}+{\it\epsilon}_{j}z_{2}+{\it\epsilon}_{k}z_{3}=0}}1.\end{eqnarray}$$
Hence we can replace
$z_{1},z_{2}$
and
$z_{3}$
by linear functions of
$y$
and
$w$
, and it follows that
$$\begin{eqnarray}{\mathcal{J}}_{4}\ll X^{5}L^{9}.\end{eqnarray}$$
Hence we can obtain again that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/3}.\end{eqnarray}$$
The proof of Lemma 5.3 is finished.
5.3 Proof of Lemma 5.4
The proof of Lemma 5.10 can be modified to establish the following result. The detail of the proof is omitted.
Lemma 5.11. If
$\text{rank}(B_{1})=2$
and
$\text{rank}(B_{2})=\text{rank}(B_{3})=3$
, then we can write
$A$
in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}A_{1} & {\it\gamma}_{1} & ({\it\gamma}_{2},{\it\gamma}_{3})C\\ {\it\gamma}_{1}^{T} & a & {\it\upsilon}^{T}\\ C^{T}({\it\gamma}_{2},{\it\gamma}_{3})^{T} & {\it\upsilon} & D+C^{T}HC\end{array}\right),\end{eqnarray}$$
where
${\it\gamma}_{1}\in \mathbb{Z}^{3}$
,
${\it\gamma}_{2},{\it\gamma}_{3}\in \mathbb{Q}^{3}$
,
$C\in M_{2,n-4}(\mathbb{Z})$
,
$a\in \mathbb{Z}$
,
${\it\upsilon}\in \mathbb{Z}^{n-4}$
,
$H\in M_{2,2}(\mathbb{Q})$
and
$D=\text{diag}\{d_{1},\ldots ,d_{n-4}\}\in M_{n-4,n-4}(\mathbb{Q})$
is a diagonal matrix. Moreover, one has
$({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\in GL_{3}(\mathbb{Q})$
.
Lemma 5.12. Let
$A$
be given by (5.26). We write
$$\begin{eqnarray}C=({\it\xi}_{1},\ldots ,{\it\xi}_{n-4})\quad \text{and}\quad {\it\upsilon}^{T}=(v_{1},\ldots ,v_{n-4}).\end{eqnarray}$$
Let
$$\begin{eqnarray}R_{i,j,k}=\left(\begin{array}{@{}ccc@{}}{\it\xi}_{i} & {\it\xi}_{j} & {\it\xi}_{k}\\ v_{i} & v_{j} & v_{k}\end{array}\right).\end{eqnarray}$$
Under the conditions in Lemma 5.11, one can find pairwise distinct
$i,j,k,u$
with
$1\leqslant i,j,k,u\leqslant n-4$
such that at least one of the following two statements holds: (i)
$\text{rank}(R_{i,j,k})=3$
and
$d_{u}\not =0$
; (ii)
$\text{rank}({\it\xi}_{i},{\it\xi}_{j})=2$
and
$d_{k}d_{u}\not =0$
.
Proof. It follows from
$9\leqslant \text{rank}(A)\leqslant \text{rank}(D)+\text{rank}({\it\upsilon})+\text{rank}(C)+4$
that
$\text{rank}(D)\geqslant 2$
. If
$\text{rank}(D)=2$
, say
$d_{1}d_{2}\not =0$
, then
$\text{rank}(R)\geqslant 3$
, where
$$\begin{eqnarray}R=\left(\begin{array}{@{}ccc@{}}{\it\xi}_{3} & \cdots \, & {\it\xi}_{n-4}\\ v_{3} & \cdots \, & v_{n-4}\end{array}\right).\end{eqnarray}$$
Then statement (i) holds. Next we assume
$\text{rank}(D)\geqslant 3$
. Note that
$\text{rank}({\it\xi}_{1},{\it\xi}_{2})=2$
due to
$\text{rank}(B)=3$
. If
$d_{r}d_{s}\not =0$
for some
$r>s\geqslant 3$
, then statement (ii) follows by choosing
$i=1,j=2,k=r$
and
$u=s$
. Therefore, we now assume that
$\text{rank}(D)=3$
and
$d_{1}d_{2}\not =0$
. Without loss of generality, we suppose that
$d_{3}\not =0$
and
$d_{s}=0(4\leqslant s\leqslant n-4)$
. We consider
$\text{rank}({\it\xi}_{1},{\it\xi}_{s})$
and
$\text{rank}({\it\xi}_{2},{\it\xi}_{s})$
for
$4\leqslant s\leqslant n-4$
. If
$\text{rank}({\it\xi}_{1},{\it\xi}_{s})=2$
for some
$s$
with
$4\leqslant s\leqslant n-4$
, then one can choose
$i=1$
,
$j=s$
$k=2$
and
$u=3$
to establish statement (ii). Similarly, statement (ii) follows if
$\text{rank}({\it\xi}_{2},{\it\xi}_{s})=2$
for some
$s$
with
$4\leqslant s\leqslant n-4$
. Thus it remains to consider the case
$\text{rank}({\it\xi}_{1},{\it\xi}_{s})=\text{rank}({\it\xi}_{2},{\it\xi}_{s})=1$
for
$4\leqslant s\leqslant n-4$
. However, it follows from
$\text{rank}({\it\xi}_{1},{\it\xi}_{2})=\text{rank}({\it\xi}_{1},{\it\xi}_{2},{\it\xi}_{s})=2$
that
${\it\xi}_{s}=0$
, and this is contradictory to the condition
$\text{rank}(A)\geqslant 9$
. We complete the proof of Lemma 5.12.◻
Proof of Lemma 5.4.
We deduce from Lemma 5.11 that
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ 1\leqslant y\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-4} \\ 1\leqslant \mathbf{z}\leqslant X}}\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}({\it\gamma}_{2},{\it\gamma}_{3})C\mathbf{z}+\mathbf{z}^{T}D\mathbf{z}+\mathbf{z}^{T}C^{T}HC\mathbf{z}))\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(2\mathbf{x}^{T}{\it\gamma}_{1}y+ay^{2}+2\mathbf{z}^{T}{\it\upsilon}y)){\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z}).\nonumber\end{eqnarray}$$
We introduce new variables
$w\in \mathbb{Z}$
and
$\mathbf{h}\in \mathbb{Z}^{2}$
to replace
$2\mathbf{x}^{T}{\it\gamma}_{1}+ay+2\mathbf{z}^{T}{\it\upsilon}^{T}$
and
$C\mathbf{z}$
, respectively. Therefore, we have
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \int _{[0,1]^{3}}\mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{Z}^{2} \\ |\mathbf{h}|\ll X}}\mathop{\sum }_{\substack{ |w|\ll X}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{1\leqslant y\leqslant X}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{N}^{n-4} \\ 1\leqslant \mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{z})\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}({\it\gamma}_{2},{\it\gamma}_{3})\mathbf{h}+\mathbf{z}^{T}D\mathbf{z}+\mathbf{h}^{T}H\mathbf{h}+wy))\nonumber\\ \displaystyle & & \displaystyle \times \,e((2\mathbf{x}^{T}{\it\gamma}_{1}+ay+2\mathbf{z}^{T}{\it\upsilon}-w){\it\beta}_{1})\nonumber\\ \displaystyle & & \displaystyle \times \,e((C\mathbf{z}-\mathbf{h})^{T}{\bf\beta}^{\prime })\,d{\bf\beta},\nonumber\end{eqnarray}$$
where
${\bf\beta}=({\it\beta}_{1},{\it\beta}_{2},{\it\beta}_{3})^{T}$
,
${\bf\beta}^{\prime }=({\it\beta}_{2},{\it\beta}_{3})^{T}$
and
$d{\bf\beta}=d{\it\beta}_{1}\,d{\it\beta}_{2}\,d{\it\beta}_{3}$
. Now we introduce
$$\begin{eqnarray}\displaystyle {\mathcal{F}}({\it\alpha},{\bf\beta}) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{h}\in \mathbb{Z}^{2} \\ |\mathbf{h}|\ll X}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}({\it\gamma}_{2},{\it\gamma}_{3})\mathbf{h}+\mathbf{h}^{T}H\mathbf{h}))\nonumber\\ \displaystyle & & \displaystyle \times \,e(2\mathbf{x}^{T}{\it\gamma}_{1}{\it\beta}_{1}-\mathbf{h}^{T}{\bf\beta}^{\prime }){\rm\Lambda}(\mathbf{x}),\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{H}}({\it\alpha},{\bf\beta})=\mathop{\sum }_{|w|\ll X}\mathop{\sum }_{\substack{ 1\leqslant y\leqslant X}}e({\it\alpha}wy+(ay-w){\it\beta}_{1}){\rm\Lambda}(y).\end{eqnarray}$$
On recalling notations in (5.27), we define
$$\begin{eqnarray}f_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{\substack{ 1\leqslant z\leqslant X}}e({\it\alpha}d_{j}z^{2}+2zv_{j}{\it\beta}_{1}+z{\it\xi}_{j}^{T}{\bf\beta}^{\prime }){\rm\Lambda}(z).\end{eqnarray}$$
Then we obtain from above
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant \int _{\mathfrak{m}}\int _{[0,1]^{3}}\bigg|{\mathcal{F}}({\it\alpha},{\bf\beta}){\mathcal{H}}({\it\alpha},{\bf\beta})\mathop{\prod }_{j=1}^{n-4}f_{j}({\it\alpha},{\bf\beta})\bigg|\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
One can deduce from Lemmas 5.6 and 5.7 that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[0,1]^{4}}|{\mathcal{H}}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})f_{k}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{8}\mathop{\sum }_{\substack{ |w|,|w^{\prime }|,|y|,|y^{\prime }|,|z_{i}|,|z_{i}^{\prime }|,|z_{j}|,|z_{j}^{\prime }|,|z_{k}|,|z_{k}^{\prime }|\ll X \\ wy^{\prime }+yw^{\prime }+2(d_{i}z_{i}z_{i}^{\prime }+d_{j}z_{j}z_{j}^{\prime }+d_{k}z_{k}z_{k}^{\prime })=0 \\ ay-w+2(v_{i}z_{i}+v_{j}z_{j}+v_{k}z_{k})=0 \\ z_{i}{\it\xi}_{i}+z_{j}{\it\xi}_{j}+z_{k}{\it\xi}_{k}=0}}1.\nonumber\end{eqnarray}$$
If
$\text{rank}(R_{i,j,k})=3$
, then we can represent
$z_{i},z_{j}$
and
$z_{k}$
by linear functions of
$y$
and
$w$
. Then by Lemma 5.8,
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{H}}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})f_{k}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{9}.\end{eqnarray}$$
If
$\text{rank}({\it\xi}_{i},{\it\xi}_{j})=2$
, then we can represent
$z_{i},z_{j}$
and
$w$
by linear functions of
$y$
and
$z_{k}$
. Then we obtain by Lemma 5.8 again
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{H}}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})f_{k}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{9}\end{eqnarray}$$
provided that
$d_{k}\not =0$
. By Lemmas 5.6–5.7, we can obtain
$$\begin{eqnarray}\displaystyle \int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha} & \ll & \displaystyle L^{6}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|\mathbf{h}|,|\mathbf{h}^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{x}^{T}({\it\gamma}_{2},{\it\gamma}_{3})\mathbf{h}^{\prime }+{\mathbf{x}^{\prime }}^{T}({\it\gamma}_{2},{\it\gamma}_{3})\mathbf{h}+\mathbf{h}^{T}H\mathbf{h}^{\prime }=0 \\ \mathbf{x}^{T}{\it\gamma}_{1}=0 \\ \mathbf{h}^{T}=0}}1\nonumber\\ \displaystyle & = & \displaystyle L^{6}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|\mathbf{h}^{\prime }|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{x}^{T}({\it\gamma}_{2},{\it\gamma}_{3})\mathbf{h}^{\prime }=0 \\ \mathbf{x}^{T}{\it\gamma}_{1}=0}}1.\nonumber\end{eqnarray}$$
Then we deduce that
$$\begin{eqnarray}\displaystyle \int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha} & \ll & \displaystyle L^{6}\mathop{\sum }_{\substack{ |\mathbf{x}|,|\mathbf{x}^{\prime }|,|\mathbf{h}^{\prime }|,|\mathbf{h}|\ll X \\ \mathbf{x}^{T}A_{1}\mathbf{x}^{\prime }+\mathbf{h}^{T}\mathbf{h}^{\prime }=0 \\ \mathbf{x}^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})=(0,\mathbf{h}^{T})}}1\nonumber\\ \displaystyle & \ll & \displaystyle L^{6}\mathop{\sum }_{\substack{ |\mathbf{x}^{\prime }|,|\mathbf{h}^{\prime }|,|\mathbf{h}|\ll X \\ (0,\mathbf{h}^{T})({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})^{-1}A_{1}\mathbf{x}^{\prime }+\mathbf{h}^{T}\mathbf{h}^{\prime }=0}}1.\nonumber\end{eqnarray}$$
On invoking Lemma 5.8, we arrive at
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{7}.\end{eqnarray}$$
If
$1\leqslant i,j,k\leqslant n-4$
are pairwise distinct, then one has by (5.29) and the Cauchy–Schwarz inequality
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha} & {\leqslant} & \displaystyle \sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{u\not =i,j,k}f_{u}({\it\alpha},{\bf\beta})\bigg|\,\bigg(\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg(\int _{[0,1]^{4}}|{\mathcal{H}}({\it\alpha},{\bf\beta})f_{i}({\it\alpha},{\bf\beta})f_{j}({\it\alpha},{\bf\beta})f_{k}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}.\nonumber\end{eqnarray}$$
Now it follows from above together with Lemmas 4.3 and 5.12 that
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
We complete the proof of Lemma 5.4.
5.4 Proof of Lemma 5.5
Similar to Lemmas 5.9–5.11, we also have the following result.
Lemma 5.13. If
$\text{rank}(B_{1})=\text{rank}(B_{2})=\text{rank}(B_{3})=3$
, then we can write
$A$
in the form
$$\begin{eqnarray}A=\left(\begin{array}{@{}cc@{}}A_{1} & ({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})C\\ C^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})^{T} & D+C^{T}HC\end{array}\right),\end{eqnarray}$$
where
$C\in M_{3,n-3}(\mathbb{Z})$
,
${\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3}\in \mathbb{Q}^{3}$
,
$H\in M_{3,3}(\mathbb{Q})$
and
$D=\text{diag}\{d_{1},\ldots ,d_{n-3}\}\in M_{n-3,n-3}(\mathbb{Q})$
is a diagonal matrix. Furthermore, we have
$({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\in GL_{3}(\mathbb{Q})$
.
Lemma 5.14. Let
$A$
be given by (5.30) satisfying the conditions in Lemma 5.13. We write
$$\begin{eqnarray}C=({\it\xi}_{1},\ldots ,{\it\xi}_{n-3}).\end{eqnarray}$$
Then we can find pairwise distinct
$u_{j}(1\leqslant j\leqslant 6)$
with
$1\leqslant u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}\leqslant n-3$
so that
$\text{rank}({\it\xi}_{u_{1}},{\it\xi}_{u_{2}},{\it\xi}_{u_{3}})=3$
and
$d_{u_{4}}d_{u_{5}}d_{u_{6}}\not =0$
.
Proof. It follows from
$\text{rank}(A)\geqslant 9$
that
$\text{rank}(D)\geqslant 3$
. If
$\text{rank}(D)=3$
, then we may assume that
$d_{1}d_{2}d_{3}\not =0$
and
$d_{j}=0$
for
$j\geqslant 4$
. Thus
$\text{rank}({\it\xi}_{4},\ldots ,{\it\xi}_{n-3})=3$
, and the desired conclusion follows. Next we assume
$\text{rank}(D)\geqslant 4$
. Since
$\text{rank}({\it\xi}_{1},{\it\xi}_{2},{\it\xi}_{3})=3$
, the desired conclusion follows again if there are distinct
$k_{1},k_{2}$
and
$k_{3}$
such that
$d_{k_{1}}d_{k_{2}}d_{k_{3}}\not =0$
and
$k_{1},k_{2},k_{3}\geqslant 4$
. Thus we now assume that for any distinct
$k_{1},k_{2},k_{3}\geqslant 4$
, one has
$d_{k_{1}}d_{k_{2}}d_{k_{3}}=0$
. This yields
$\text{rank}(D)\leqslant 5$
. We first consider the case
$\text{rank}(D)=4$
. There are at least two distinct
$j_{1},j_{2}\leqslant 3$
such that
$d_{j_{1}}d_{j_{2}}\not =0$
. Suppose that
$d_{s_{i}}=0$
for
$1\leqslant i\leqslant n-7$
. Then the rank of
$\{{\it\xi}_{s_{i}}\}_{1\leqslant i\leqslant n-7}$
is at least 2, say
$\text{rank}({\it\xi}_{s_{1}},{\it\xi}_{s_{2}})=2$
. Since
$\text{rank}({\it\xi}_{1},{\it\xi}_{2},{\it\xi}_{3})=3$
, we can find
$j$
with
$1\leqslant j\leqslant 3$
such that
$\text{rank}({\it\xi}_{j},{\it\xi}_{s_{1}},{\it\xi}_{s_{2}})=3$
. The desired conclusion follows easily by choosing
$u_{1}=j$
,
$u_{2}=s_{1}$
and
$u_{3}=s_{2}$
. Now we consider the case
$\text{rank}(D)=5$
, and we may assume that
$d_{1}d_{2}d_{3}d_{4}d_{5}\not =0$
and
$d_{r}=0$
for
$r\geqslant 6$
. Since
$\text{rank}(A)\geqslant 9$
, there exist
$r\geqslant 6$
(say
$r=6$
) such that
${\it\xi}_{r}\not =0$
. Then one can choose
$j_{1},j_{2}\leqslant 3$
so that
$\text{rank}({\it\xi}_{j_{1}},{\it\xi}_{j_{2}},{\it\xi}_{6})=3$
. The desired conclusion follows by choosing
$u_{1}=j_{1}$
,
$u_{2}=j_{2}$
and
$u_{3}=6$
. The proof of Lemma 5.14 is completed.◻
Proof of Lemma 5.5.
We apply Lemma 5.13 to conclude that
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{n-3} \\ 1\leqslant \mathbf{y}\leqslant X}}{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+\mathbf{y}^{T}D\mathbf{y}))\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(2\mathbf{x}^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})C\mathbf{y}+\mathbf{y}^{T}C^{T}HC\mathbf{y})).\nonumber\end{eqnarray}$$
By orthogonality, one has
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \int _{[0,1]^{3}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{y}\in \mathbb{N}^{n-3} \\ 1\leqslant \mathbf{y}\leqslant X}}\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{Z}^{3} \\ |\mathbf{z}|\ll X}}{\rm\Lambda}(\mathbf{x}){\rm\Lambda}(\mathbf{y})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+\mathbf{y}^{T}D\mathbf{y}))\nonumber\\ \displaystyle & & \displaystyle \times \,e({\it\alpha}(2\mathbf{x}^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\mathbf{z}+\mathbf{z}^{T}H\mathbf{z}))\nonumber\\ \displaystyle & & \displaystyle \times \,e((\mathbf{y}^{T}C^{T}-\mathbf{z}^{T}){\bf\beta})\,d{\bf\beta},\nonumber\end{eqnarray}$$
where
${\bf\beta}=({\it\beta}_{1},{\it\beta}_{2},{\it\beta}_{3})^{T}$
and
$d{\bf\beta}=d{\it\beta}_{1}\,d{\it\beta}_{2}\,d{\it\beta}_{3}$
. Now we introduce
$$\begin{eqnarray}{\mathcal{F}}({\it\alpha},{\bf\beta})\,=\!\mathop{\sum }_{\substack{ \mathbf{x}\in \mathbb{N}^{3} \\ 1\leqslant \mathbf{x}\leqslant X}}\!\mathop{\sum }_{\substack{ \mathbf{z}\in \mathbb{Z}^{3} \\ 1\leqslant \mathbf{z}\leqslant X}}\!{\rm\Lambda}(\mathbf{x})e({\it\alpha}(\mathbf{x}^{T}A_{1}\mathbf{x}+2\mathbf{x}^{T}({\it\gamma}_{1},{\it\gamma}_{2},{\it\gamma}_{3})\mathbf{z}+\mathbf{z}^{T}H\mathbf{z})-\mathbf{z}^{T}{\bf\beta}),\end{eqnarray}$$
and
$$\begin{eqnarray}f_{j}({\it\alpha},{\bf\beta})=\mathop{\sum }_{\substack{ 1\leqslant y\leqslant X}}e(d_{j}{\it\alpha}y^{2}+2y{\it\xi}_{j}^{T}{\bf\beta}){\rm\Lambda}(y),\end{eqnarray}$$
where
${\it\xi}_{1},\ldots ,{\it\xi}_{n-3}$
is given by (5.31). We conclude from above
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\leqslant \int _{\mathfrak{m}}\int _{[0,1]^{3}}\bigg|{\mathcal{F}}({\it\alpha},{\bf\beta})\mathop{\prod }_{j=1}^{n-3}f_{j}({\it\alpha},{\bf\beta})\bigg|\,d{\bf\beta}\,d{\it\alpha}.\end{eqnarray}$$
One applying Lemmas 5.6–5.8, we can easily establish
$$\begin{eqnarray}\int _{[0,1]^{4}}\bigg|\mathop{\prod }_{i=1}^{5}f_{u_{i}}({\it\alpha},{\bf\beta})\bigg|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{5}L^{11}\end{eqnarray}$$
provided that
$\text{rank}({\it\xi}_{u_{1}},{\it\xi}_{u_{2}},{\it\xi}_{u_{3}})=3$
and
$d_{u_{4}}d_{u_{5}}\not =0$
. Similarly, we also have
$$\begin{eqnarray}\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\ll X^{7}L^{7}.\end{eqnarray}$$
By (5.32) and the Cauchy–Schwarz inequality, one has for distinct
$u_{1},u_{2},u_{3},u_{4}$
and
$u_{5}$
that
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha} & {\leqslant} & \displaystyle \sup _{\substack{ {\it\alpha}\in \mathfrak{m} \\ {\bf\beta}\in [0,1]^{3}}}\bigg|\mathop{\prod }_{k\not =u_{1},u_{2},u_{3},u_{4},u_{5}}f_{k}({\it\alpha},{\bf\beta})\bigg|\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg(\int _{[0,1]^{4}}|{\mathcal{F}}({\it\alpha},{\bf\beta})|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\bigg(\int _{[0,1]^{4}}\bigg|\mathop{\prod }_{i=1}^{5}f_{u_{i}}({\it\alpha},{\bf\beta})\bigg|^{2}\,d{\bf\beta}\,d{\it\alpha}\bigg)^{1/2}.\end{eqnarray}$$
Combining (5.33)–(5.35), Lemma 4.3 and Lemma 5.14, one has
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/6}.\end{eqnarray}$$
The proof of Lemma 5.5 is finished.
6 Quadratic forms with off-diagonal rank
${\geqslant}4$
Proposition 6.1. Let
$A$
be defined in (1.1), and let
$S({\it\alpha})$
be defined in (2.5). We write
$$\begin{eqnarray}G=\left(\begin{array}{@{}ccc@{}}a_{1,5} & \cdots \, & a_{1,9}\\ \vdots & \cdots \, & \vdots \\ a_{5,5} & \cdots \, & a_{5,9}\end{array}\right).\end{eqnarray}$$
Suppose that
$\det (G)\not =0$
. Then we have
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/20},\end{eqnarray}$$
where the implied constant depends on
$A$
and
$K$
.
Throughout this section, we shall assume that the matrix
$G$
given by (6.1) is invertible.
Lemma 6.2. Let
${\it\tau}\not =0$
be a real number. Then we have
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\mathop{\sum }_{|x|\ll X}\min \{X,\Vert x{\it\tau}({\it\alpha}-{\it\beta})\Vert ^{-1}\}\,d{\it\alpha}\,d{\it\beta}\ll LQ^{7/2}X^{-2},\end{eqnarray}$$
where the implied constant depends on
${\it\tau}$
.
Proof. Without loss of generality, we assume that
$0<|{\it\tau}|\leqslant 1$
. Thus
$|{\it\tau}({\it\alpha}-{\it\beta})|\leqslant 1$
. We introduce
$$\begin{eqnarray}{\mathcal{M}}=\mathop{\bigcup }_{1\leqslant q\leqslant Q^{1/2}}\mathop{\bigcup }_{\substack{ -q\leqslant a\leqslant q \\ (a,q)=1}}\biggl\{\big|{\it\alpha}-\frac{a}{q}\big|\leqslant \frac{Q^{1/2}}{qX^{2}}\biggr\}.\end{eqnarray}$$
By Dirichlet’s approximation theorem, there exist
$a\in \mathbb{Z}$
and
$q\in \mathbb{N}$
with
$(a,q)=1$
,
$1\leqslant q\leqslant X^{2}Q^{-1/2}$
and
$|{\it\tau}({\it\alpha}-{\it\beta})-a/q|\leqslant Q^{1/2}(qX^{2})^{-1}$
. Since
$|{\it\tau}({\it\alpha}-{\it\beta})|\leqslant 1$
, one has
$-q\leqslant a\leqslant q$
. If
${\it\tau}({\it\alpha}-{\it\beta})\not \in {\mathcal{M}}$
, then
$q>Q^{1/2}$
. By Vaughan [Reference Vaughan12, Lemma 2.2],
$$\begin{eqnarray}\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\tau}({\it\alpha}-{\it\beta})\Vert ^{-1}\}\ll LQ^{-1/2}X^{2}.\end{eqnarray}$$
Therefore, we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathfrak{m}(Q)}\int _{\substack{ \mathfrak{m}(Q) \\ {\it\tau}({\it\alpha}-{\it\beta})\not \in {\mathcal{M}}}}\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\tau}({\it\alpha}-{\it\beta})\Vert ^{-1}\}\,d{\it\alpha}d{\it\beta}\nonumber\\ \displaystyle & & \displaystyle \quad \ll LQ^{-1/2}X^{2}\int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\,d{\it\alpha}\,d{\it\beta}\ll LQ^{7/2}X^{-2}.\nonumber\end{eqnarray}$$
When
${\it\tau}({\it\alpha}-{\it\beta})\in {\mathcal{M}}$
, we apply the trivial bound to the summation over
$x$
to deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathfrak{m}(Q)}\int _{\substack{ \mathfrak{m}(Q) \\ {\it\tau}({\it\alpha}-{\it\beta})\in {\mathcal{M}}}}\mathop{\sum }_{|x|\ll X}\min \{X,~\Vert x{\it\tau}({\it\alpha}-{\it\beta})\Vert ^{-1}\}\,d{\it\alpha}\,d{\it\beta}\nonumber\\ \displaystyle & & \displaystyle \quad \ll X^{2}\int _{\mathfrak{m}(Q)}\int _{\substack{ \mathfrak{m}(Q) \\ {\it\tau}({\it\alpha}-{\it\beta})\in {\mathcal{M}}}}\,d{\it\alpha}\,d{\it\beta}\ll X^{2}(Q^{2}X^{-2}QX^{-2})=Q^{3}X^{-2}.\nonumber\end{eqnarray}$$
The desired conclusion follows from above immediately. ◻
To introduce the next lemma, we define
$$\begin{eqnarray}{\rm\Phi}({\it\alpha})=\min \{X,~\Vert {\it\alpha}\Vert ^{-1}\}.\end{eqnarray}$$
For
$\mathbf{v}=(v_{1},\ldots ,v_{5})\in \mathbb{Z}^{5}$
and
$G$
given by (6.1), we write
$$\begin{eqnarray}2G\mathbf{v}=\left(\begin{array}{@{}c@{}}g_{1}(\mathbf{v})\\ \vdots \\ g_{5}(\mathbf{v})\end{array}\right).\end{eqnarray}$$
Lemma 6.3. One has
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}|S({\it\alpha})|^{2}\,d{\it\alpha}\ll X^{2n-10}L^{2n-6}\int _{0}^{1}\bigg(\int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\bigg){\rm\Phi}({\it\gamma})d{\it\gamma},\end{eqnarray}$$
where
$$\begin{eqnarray}J_{{\it\gamma}}({\it\alpha})=\mathop{\sum }_{|\mathbf{v}|\leqslant X}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+v_{1})e({\it\alpha}zg_{5}(\mathbf{v}))e({\it\gamma}z)\bigg|\mathop{\prod }_{j=1}^{4}{\rm\Phi}(g_{j}(\mathbf{v}){\it\alpha}).\end{eqnarray}$$
Proof. Let
$$\begin{eqnarray}\displaystyle r(\mathbf{y}) & = & \displaystyle \mathop{\sum }_{i=1}^{4}\mathop{\sum }_{j=1}^{4}a_{i,j}y_{i}y_{j},\quad q(\mathbf{z})=\mathop{\sum }_{i=5}^{9}\mathop{\sum }_{j=5}^{9}a_{i,j}z_{i}z_{j}\quad \text{and}\quad \nonumber\\ \displaystyle p(\mathbf{w}) & = & \displaystyle \mathop{\sum }_{i=10}^{n}\mathop{\sum }_{j=10}^{n}a_{i,j}w_{i}w_{j}.\nonumber\end{eqnarray}$$
We set
$$\begin{eqnarray}B=(2a_{i,j})_{\substack{ 1\leqslant i\leqslant 4,10\leqslant j\leqslant n}}\quad \text{and}\quad C=(2a_{i,j})_{5\leqslant i\leqslant 9,10\leqslant j\leqslant n}.\end{eqnarray}$$
Then
$f$
can be written in the form
$$\begin{eqnarray}f(\mathbf{x})=r(\mathbf{y})+y_{1}g_{1}(\mathbf{z})+\cdots +y_{4}g_{4}(\mathbf{z})+q(\mathbf{z})+\mathbf{y}^{T}B\mathbf{w}+\mathbf{z}^{T}C\mathbf{w}+p(\mathbf{w}),\end{eqnarray}$$
where
$\mathbf{z}=(z_{1},\ldots ,z_{5})$
,
$\mathbf{y}=(y_{1},\ldots ,y_{4})$
,
$\mathbf{w}=(w_{1},\ldots ,w_{n-9})$
. Note that
$\mathbf{y}^{T}B\mathbf{w}+\mathbf{z}^{T}C\mathbf{w}+p(\mathbf{w})$
vanishes if
$n=9$
. Therefore, one has
$$\begin{eqnarray}\displaystyle S({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{w}\leqslant X}}\mathop{\sum }_{1\leqslant \mathbf{z}\leqslant X}{\rm\Lambda}(\mathbf{z})e({\it\alpha}(y_{1}g_{1}(\mathbf{z})+\cdots +y_{4}g_{4}(\mathbf{z})+q(\mathbf{z})+\mathbf{z}^{T}C\mathbf{w}))\nonumber\\ \displaystyle & & \displaystyle \times \,{\rm\Lambda}(\mathbf{y}){\rm\Lambda}(\mathbf{w})e({\it\alpha}(r(\mathbf{y})+\mathbf{y}^{T}B\mathbf{w}+p(\mathbf{w}))).\nonumber\end{eqnarray}$$
By Cauchy’s inequality,
$$\begin{eqnarray}|S({\it\alpha})|^{2}\leqslant X^{n-5}L^{2n-10}T({\it\alpha}),\end{eqnarray}$$
where
$$\begin{eqnarray}T({\it\alpha})=\mathop{\sum }_{\substack{ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{w}\leqslant X}}\bigg|\mathop{\sum }_{1\leqslant \mathbf{z}\leqslant X}{\rm\Lambda}(\mathbf{z})e\bigg({\it\alpha}\bigg(\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{z})+q(\mathbf{z})+\mathbf{z}^{T}C\mathbf{w}\bigg)\bigg)\bigg|^{2}.\end{eqnarray}$$
Then we deduce that
$$\begin{eqnarray}\displaystyle T({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{w}\leqslant X}}\mathop{\sum }_{1\leqslant \mathbf{z}_{1}\leqslant X}\mathop{\sum }_{1\leqslant \mathbf{z}_{2}\leqslant X}{\rm\Lambda}(\mathbf{z}_{1}){\rm\Lambda}(\mathbf{z}_{2})\nonumber\\ \displaystyle & & \displaystyle \times \,e\bigg({\it\alpha}\bigg(\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{z}_{1}-\mathbf{z}_{2})+q(\mathbf{z}_{1})-q(\mathbf{z}_{2})\bigg)\bigg)e({\it\alpha}(\mathbf{z}_{1}-\mathbf{z}_{2})^{T}C\mathbf{w})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{1\leqslant \mathbf{z}_{1}\leqslant X}\mathop{\sum }_{1\leqslant \mathbf{z}_{2}\leqslant X}{\rm\Lambda}(\mathbf{z}_{1}){\rm\Lambda}(\mathbf{z}_{2})\mathop{\sum }_{\substack{ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{w}\leqslant X}}\nonumber\\ \displaystyle & & \displaystyle \times \,e\bigg({\it\alpha}\bigg(\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{z}_{1}-\mathbf{z}_{2})+q(\mathbf{z}_{1})-q(\mathbf{z}_{2})\bigg)\bigg)e({\it\alpha}(\mathbf{z}_{1}-\mathbf{z}_{2})^{T}C\mathbf{w}).\nonumber\end{eqnarray}$$
By changing variables
$\mathbf{z}_{1}=\mathbf{z}_{2}+\mathbf{v}$
, we have
$$\begin{eqnarray}\displaystyle T({\it\alpha}) & = & \displaystyle \mathop{\sum }_{1\leqslant \mathbf{z}\leqslant X}\mathop{\sum }_{\substack{ |\mathbf{v}|\leqslant X \\ 1\leqslant \mathbf{v}+\mathbf{z}\leqslant X}}{\rm\Lambda}(\mathbf{z}){\rm\Lambda}(\mathbf{z}+\mathbf{v})\mathop{\sum }_{\substack{ 1\leqslant \mathbf{y}\leqslant X \\ 1\leqslant \mathbf{w}\leqslant X}}\nonumber\\ \displaystyle & & \displaystyle \times \,e\bigg({\it\alpha}\bigg(\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{v})+q(\mathbf{z}+\mathbf{v})-q(\mathbf{z})\bigg)\bigg)e({\it\alpha}\mathbf{v}^{T}C\mathbf{w}).\nonumber\end{eqnarray}$$
We exchange the summation over
$\mathbf{z}$
and the summation over
$\mathbf{v}$
to obtain
$$\begin{eqnarray}T({\it\alpha})=\mathop{\sum }_{|\mathbf{v}|\leqslant X}\bigg(\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e\bigg({\it\alpha}\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{v})\bigg)\bigg)R(\mathbf{v})\mathop{\prod }_{j=1}^{5}{\mathcal{K}}_{j,\mathbf{v}}({\it\alpha}),\end{eqnarray}$$
where
$$\begin{eqnarray}R(\mathbf{v})=e({\it\alpha}q(\mathbf{v}))\mathop{\sum }_{\substack{ 1\leqslant \mathbf{w}\leqslant X}}e({\it\alpha}(\mathbf{v}^{T}C\mathbf{w}))\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{K}}_{j,\mathbf{v}}({\it\alpha})=\mathop{\sum }_{\substack{ 1\leqslant z_{j}\leqslant X \\ 1-v_{j}\leqslant z_{j}\leqslant X-v_{j}}}{\rm\Lambda}(z_{j}){\rm\Lambda}(z_{j}+v_{j})e\bigg(2{\it\alpha}z_{j}\mathop{\sum }_{k=1}^{5}a_{j+4,k+4}v_{k}\bigg).\end{eqnarray}$$
The range of
$z_{j}$
in summation (6.7) depends on
$v_{j}$
. We first follow the standard argument (see for example the argument around (15) in [Reference Wooley15]) to remove the dependence on
$v_{j}$
. We write
$$\begin{eqnarray}{\mathcal{G}}_{v_{1}}({\it\gamma})=\mathop{\sum }_{\substack{ 1\leqslant z\leqslant X \\ 1-v_{1}\leqslant z\leqslant X-v_{1}}}e(-z{\it\gamma})\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma})=\mathop{\sum }_{\substack{ |z|\leqslant X}}{\rm\Lambda}(z){\rm\Lambda}(z+v_{1})e({\it\alpha}zg_{5}(\mathbf{v}))e({\it\gamma}z).\end{eqnarray}$$
Then we deduce from (6.7)–(6.9) that
$$\begin{eqnarray}{\mathcal{K}}_{1,\mathbf{v}}({\it\alpha})=\int _{0}^{1}{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma}){\mathcal{G}}_{v_{1}}({\it\gamma})\,d{\it\gamma}.\end{eqnarray}$$
On substituting (6.10) into (6.6), we obtain
$$\begin{eqnarray}\displaystyle T({\it\alpha}) & = & \displaystyle \mathop{\sum }_{|\mathbf{v}|\leqslant X}R(\mathbf{v})\mathop{\prod }_{j=2}^{5}{\mathcal{K}}_{j,\mathbf{v}}({\it\alpha})\bigg(\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e\bigg({\it\alpha}\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{v})\bigg)\bigg)\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{0}^{1}{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma}){\mathcal{G}}_{v_{1}}({\it\gamma})\,d{\it\gamma}\nonumber\\ \displaystyle & = & \displaystyle \int _{0}^{1}\mathop{\sum }_{|\mathbf{v}|\leqslant X}R(\mathbf{v})\mathop{\prod }_{j=2}^{5}{\mathcal{K}}_{j,\mathbf{v}}({\it\alpha})\bigg(\mathop{\sum }_{1\leqslant \mathbf{y}\leqslant X}e\bigg({\it\alpha}\mathop{\sum }_{j=1}^{4}y_{j}g_{j}(\mathbf{v})\bigg)\bigg)\nonumber\\ \displaystyle & & \displaystyle \times \,{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma}){\mathcal{G}}_{v_{1}}({\it\gamma})d{\it\gamma}.\nonumber\end{eqnarray}$$
Then we conclude that
$$\begin{eqnarray}|T({\it\alpha})|\ll X^{n-5}L^{4}\int _{0}^{1}\mathop{\sum }_{|\mathbf{v}|\leqslant X}|{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma})|\mathop{\prod }_{j=1}^{4}{\rm\Phi}(g_{j}(\mathbf{v}){\it\alpha}){\rm\Phi}({\it\gamma})\,d{\it\gamma}.\end{eqnarray}$$
By putting (6.11) into (6.5), one has
$$\begin{eqnarray}|S({\it\alpha})|^{2}\ll X^{2n-10}L^{2n-6}\int _{0}^{1}\mathop{\sum }_{|\mathbf{v}|\leqslant X}|{\mathcal{K}}_{0,\mathbf{v}}({\it\alpha},{\it\gamma})|\mathop{\prod }_{j=1}^{4}{\rm\Phi}(g_{j}(\mathbf{v}){\it\alpha}){\rm\Phi}({\it\gamma})\,d{\it\gamma}.\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}|S({\it\alpha})|^{2}\,d{\it\alpha}\ll X^{2n-10}L^{2n-6}\int _{0}^{1}\bigg(\int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\bigg){\rm\Phi}({\it\gamma})\,d{\it\gamma}.\end{eqnarray}$$
The proof is completed. ◻
Lemma 6.4. Let
$J_{{\it\gamma}}({\it\alpha})$
be defined in (6.4). Then one has uniformly for
${\it\gamma}\in [0,1]$
that
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\ll L^{25/4}Q^{-17/8}X^{8}.\end{eqnarray}$$
Proof. We deduce by changing variables
$\mathbf{h}=2G\mathbf{v}$
that
$$\begin{eqnarray}J_{{\it\gamma}}({\it\alpha})=\mathop{\sum }_{\substack{ |\mathbf{h}|\leqslant cX \\ (2G)^{-1}\mathbf{h}\in \mathbb{Z}^{5} \\ |(2G)^{-1}\mathbf{h}|\leqslant X}}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+\mathop{\sum }_{j=1}^{5}b_{j}h_{j})e({\it\alpha}zh_{5})e({\it\gamma}z)\bigg|\mathop{\prod }_{j=1}^{4}{\rm\Phi}(h_{j}{\it\alpha})\end{eqnarray}$$
for some constants
$c,b_{1},\ldots ,b_{5}$
depending only on
$G$
. We point out that
$b_{1},\ldots ,b_{5}$
are rational numbers, and we extend the domain of function
${\rm\Lambda}(x)$
by taking
${\rm\Lambda}(x)=0$
if
$x\in \mathbb{Q}\setminus \mathbb{N}$
. Then we have
$$\begin{eqnarray}\displaystyle J_{{\it\gamma}}({\it\alpha}) & {\leqslant} & \displaystyle \mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}+b_{5}h\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha}).\nonumber\end{eqnarray}$$
We first handle the easier case
$b_{5}=0$
. In this case, we can easily obtain a nontrivial estimate for the summation over
$h$
. By Cauchy’s inequality and Lemma 4.1, one has
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigg(\mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\bigg)^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \,(2cX+1)\mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll \,X^{2}L^{4}\mathop{\sum }_{|x|\ll X}\min \{X,\Vert x{\it\alpha}\Vert ^{-1}\}.\nonumber\end{eqnarray}$$
For
${\it\alpha}\in \mathfrak{m}(Q)$
, we apply Lemma 4.2 to deduce from above
$$\begin{eqnarray}\mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\ll L^{5/2}Q^{-1/2}X^{2}.\end{eqnarray}$$
Then for
${\it\alpha}\in \mathfrak{m}(Q)$
, we obtain
$$\begin{eqnarray}J_{{\it\gamma}}({\it\alpha})\ll L^{1/2}Q^{-1/2}X^{2}\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha})\ll L^{5/2}Q^{-9/2}X^{10},\end{eqnarray}$$
and thereby
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\ll L^{13/2}Q^{-5/2}X^{8}\end{eqnarray}$$
provided that
$b_{5}=0$
. From now on, we assume
$b_{5}\not =0$
. Then we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}+b_{5}h\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X \\ \frac{1}{b_{5}}(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j})\in \mathbb{Z} \\ |\frac{1}{b_{5}}(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j})|\leqslant cX}}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+k)e\bigg(\frac{{\it\alpha}}{b_{5}}z\bigg(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)\bigg)e({\it\gamma}z)\bigg|\nonumber\end{eqnarray}$$
for some constant
$c^{\prime }$
depending only on
$b_{1},\ldots ,b_{5}$
and
$c$
. Therefore, one has
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}+b_{5}h\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+k)e\bigg(\frac{{\it\alpha}}{b_{5}}z\bigg(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)\bigg)e({\it\gamma}z)\bigg|.\nonumber\end{eqnarray}$$
We apply Cauchy’s inequality to deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|h|\leqslant cX}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}\bigg(z+\mathop{\sum }_{j=1}^{4}b_{j}u_{j}+b_{5}h\bigg)e({\it\alpha}zh)e({\it\gamma}z)\bigg|\leqslant (2c^{\prime }X+1)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\bigg(\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+k)e\bigg(\frac{{\it\alpha}}{b_{5}}z\bigg(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)\bigg)e({\it\gamma}z)\bigg|^{2}\bigg)^{1/2}.\nonumber\end{eqnarray}$$
We apply Cauchy’s inequality again to obtain
$$\begin{eqnarray}J_{{\it\gamma}}({\it\alpha})\leqslant (2c^{\prime }X+1)^{1/2}{\rm\Xi}_{{\it\gamma}}({\it\alpha})^{1/2}\bigg(\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha})\bigg)^{1/2},\end{eqnarray}$$
where
${\rm\Xi}_{{\it\gamma}}({\it\alpha})$
is defined as
$$\begin{eqnarray}\displaystyle {\rm\Xi}_{{\it\gamma}}({\it\alpha}) & = & \displaystyle \mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\mathop{\sum }_{|z|\leqslant X}{\rm\Lambda}(z){\rm\Lambda}(z+k)e\bigg(\frac{{\it\alpha}}{b_{5}}z\bigg(k-\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)\bigg)e({\it\gamma}z)\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha}).\nonumber\end{eqnarray}$$
By Lemma 4.2,
$$\begin{eqnarray}J_{{\it\gamma}}({\it\alpha})\ll L^{2}Q^{-2}X^{9/2}{\rm\Xi}_{{\it\gamma}}({\it\alpha})^{1/2}.\end{eqnarray}$$
Therefore, we have
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha} & \ll & \displaystyle L^{2}Q^{-2}X^{9/2}\bigg(\int _{\mathfrak{m}(Q)}\,d{\it\alpha}\bigg)^{1/2}\bigg(\int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\bigg)^{1/2}\nonumber\\ \displaystyle & \ll & \displaystyle L^{2}Q^{-1}X^{7/2}\bigg(\int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\bigg)^{1/2}.\end{eqnarray}$$
Now it suffices to estimate
$\int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}$
. We observe
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\mathfrak{m}(Q)}\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}{\it\varpi}(z_{1},z_{2},k)e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\mathfrak{m}(Q)}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}{\it\varpi}(z_{1},z_{2},k)e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha},\nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}{\it\varpi}(z_{1},z_{2},k)={\rm\Lambda}(z_{1}){\rm\Lambda}(z_{1}+k){\rm\Lambda}(z_{2}){\rm\Lambda}(z_{2}+k)e({\it\gamma}(z_{1}-z_{2}))\end{eqnarray}$$
and
$$\begin{eqnarray}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})=e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})\mathop{\sum }_{j=1}^{4}b_{j}u_{j}\bigg)\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha}).\end{eqnarray}$$
We exchange the order of summation and integration to conclude that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}{\it\varpi}(z_{1},z_{2},k)\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{4}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|\nonumber\\ \displaystyle & & \displaystyle \quad =L^{4}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|.\nonumber\end{eqnarray}$$
Then the Cauchy–Schwarz inequality implies
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigg(\int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\bigg)^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{8}X^{3}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|^{2}.\end{eqnarray}$$
Now we apply the method developed by the author [Reference Zhao16] to deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}e\bigg(\frac{{\it\alpha}-{\it\beta}}{b_{5}}(z_{1}-z_{2})k\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}_{1}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u}_{1},z_{1},z_{2})\mathop{\sum }_{\substack{ |\mathbf{u}_{2}|\leqslant cX}}{\rm\Pi}(-{\it\beta},\mathbf{u}_{2},z_{1},z_{2})\,d{\it\alpha}\,d{\it\beta}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\bigg|\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}e\bigg(\frac{{\it\alpha}-{\it\beta}}{b_{5}}(z_{1}-z_{2})k\bigg)\bigg|\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\substack{ |\mathbf{u}_{1}|\leqslant cX}}\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}{\it\alpha})\mathop{\sum }_{\substack{ |\mathbf{u}_{2}|\leqslant cX}}\mathop{\prod }_{j=1}^{4}{\rm\Phi}(u_{j}^{\prime }{\it\beta})\,d{\it\alpha}\,d{\it\beta},\nonumber\end{eqnarray}$$
where
$\mathbf{u}_{1}=(u_{1},\ldots ,u_{4})^{T}\in \mathbb{Z}^{4}$
and
$\mathbf{u}_{2}=(u_{1}^{\prime },\ldots ,u_{4}^{\prime })^{T}\in \mathbb{Z}^{4}$
. Therefore, we obtain by Lemma 4.2
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll \int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\min \biggl\{X,\biggl\|\frac{{\it\alpha}-{\it\beta}}{b_{5}}(z_{1}-z_{2})\biggr\|^{-1}\biggr\}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(L^{4}Q^{-4}X^{8})^{2}\,d{\it\alpha}\,d{\it\beta}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{8}Q^{-8}X^{17}\int _{\mathfrak{m}(Q)}\int _{\mathfrak{m}(Q)}\mathop{\sum }_{|x|\leqslant X}\min \biggl\{X,\biggl\|\frac{{\it\alpha}-{\it\beta}}{b_{5}}x\biggr\|^{-1}\biggr\}\,d{\it\alpha}\,d{\it\beta}.\nonumber\end{eqnarray}$$
Then we conclude from Lemma 6.2 that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|z_{1}|\leqslant X}\mathop{\sum }_{|z_{2}|\leqslant X}\mathop{\sum }_{\substack{ |k|\leqslant c^{\prime }X}}\bigg|\int _{\mathfrak{m}(Q)}e\bigg(\frac{{\it\alpha}}{b_{5}}(z_{1}-z_{2})k\bigg)\mathop{\sum }_{\substack{ |\mathbf{u}|\leqslant cX}}{\rm\Pi}({\it\alpha},\mathbf{u},z_{1},z_{2})\,d{\it\alpha}\bigg|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll L^{9}Q^{-9/2}X^{15}.\end{eqnarray}$$
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}{\rm\Xi}_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\ll L^{17/2}Q^{-9/4}X^{9}.\end{eqnarray}$$
By substituting (6.16) into (6.13), we obtain
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}J_{{\it\gamma}}({\it\alpha})\,d{\it\alpha}\ll L^{25/4}Q^{-17/8}X^{8}\end{eqnarray}$$
provided that
$b_{5}\not =0$
.
We complete the proof in view of the argument around (6.12) and (6.17).◻
Lemma 6.5. One has
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}|S({\it\alpha})|\,d{\it\alpha}\ll L^{n+1}Q^{-1/16}X^{n-2}.\end{eqnarray}$$
Proof. By Cauchy’s inequality,
$$\begin{eqnarray}\displaystyle \int _{\mathfrak{m}(Q)}|S({\it\alpha})|\,d{\it\alpha} & {\leqslant} & \displaystyle \bigg(\int _{\mathfrak{m}(Q)}\,d{\it\alpha}\bigg)^{1/2}\bigg(\int _{\mathfrak{m}(Q)}|S({\it\alpha})|^{2}\,d{\it\alpha}\bigg)^{1/2}\nonumber\\ \displaystyle & \ll & \displaystyle QX^{-1}\bigg(\int _{\mathfrak{m}(Q)}|S({\it\alpha})|^{2}\,d{\it\alpha}\bigg)^{1/2}.\end{eqnarray}$$
It follows from Lemmas 6.3–6.4 that
$$\begin{eqnarray}\int _{\mathfrak{m}(Q)}|S({\it\alpha})|^{2}\,d{\it\alpha}\ll L^{2n+1}Q^{-17/8}X^{2n-2}\int _{0}^{1}{\rm\Phi}({\it\gamma})\,d{\it\gamma}\ll L^{2n+2}Q^{-17/8}X^{2n-2}.\end{eqnarray}$$
We finish Section 6 by pointing out that Proposition 6.1 follows from Lemma 6.5 by the dyadic argument.
7 The Proof of Theorem 1.1
By orthogonality, we have
$$\begin{eqnarray}N_{f,t}(X)=\int _{X^{-1}}^{1+X^{-1}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}.\end{eqnarray}$$
Recalling the definitions of
$\mathfrak{M}$
and
$\mathfrak{m}$
in (2.8) and (2.9), we have
$$\begin{eqnarray}N_{f,t}(X)=\int _{\mathfrak{M}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}+\int _{\mathfrak{m}}S({\it\alpha})e(-t{\it\alpha})\,d{\it\alpha}.\end{eqnarray}$$
In light of Lemma 3.6, to establish the asymptotic formula (1.3), it suffices to prove
$$\begin{eqnarray}\int _{\mathfrak{m}}|S({\it\alpha})|\,d{\it\alpha}\ll X^{n-2}L^{-K/20}.\end{eqnarray}$$
In view of Proposition 6.1 and the work of Liu [Reference Liu9] (see also Remark of Lemma 4.4), the estimate (7.2) holds if there exists an invertible matrix
$$\begin{eqnarray}B=\left(\begin{array}{@{}ccc@{}}a_{i_{1},j_{1}} & \cdots \, & a_{i_{5},j_{5}}\\ \vdots & \cdots \, & \vdots \\ a_{i_{5},j_{1}} & \cdots \, & a_{i_{5},j_{5}}\end{array}\right)\end{eqnarray}$$
with
$$\begin{eqnarray}|\{i_{1},\ldots ,i_{5}\}\cap \{j_{1},\ldots ,j_{5}\}|\leqslant 1.\end{eqnarray}$$
Next we assume
$\text{rank}(B)\leqslant 4$
for all
$B=(a_{i_{k},j_{l}})_{1\leqslant k,l\leqslant 5}$
satisfying
$|\{i_{1},\ldots ,i_{5}\}\cap \{j_{1},\ldots ,j_{5}\}|\leqslant 1$
. This yields
$\text{rank}_{\text{off}}(A)\leqslant 4$
. By Proposition 5.1, we can establish (7.2) again if
$\text{rank}_{\text{off}}(A)\leqslant 3$
. It remains to consider the case
$\text{rank}_{\text{off}}(A)=4$
. Without loss of generality, we assume that
$\text{rank}(C)=4$
, where
$$\begin{eqnarray}C=\left(\begin{array}{@{}cccc@{}}a_{1,5} & a_{1,6} & a_{1,7} & a_{1,8}\\ a_{2,5} & a_{2,6} & a_{2,7} & a_{2,8}\\ a_{3,5} & a_{3,6} & a_{3,7} & a_{3,8}\\ a_{4,5} & a_{4,6} & a_{4,7} & a_{4,8}\end{array}\right).\end{eqnarray}$$
Let
${\it\gamma}_{j}=(a_{j,5},\ldots ,a_{j,n})^{T}\in \mathbb{Z}^{n-4}$
for
$1\leqslant j\leqslant n$
. Then
${\it\gamma}_{1}$
,
${\it\gamma}_{2}$
,
${\it\gamma}_{3}$
and
${\it\gamma}_{4}$
are linear independent due to
$\text{rank}(C)=4$
. For
$5\leqslant k\leqslant n$
, we consider
$$\begin{eqnarray}B=\left(\begin{array}{@{}ccc@{}}a_{1,5} & \cdots \, & a_{1,n}\\ \vdots & \cdots \, & \vdots \\ a_{4,5} & \cdots \, & a_{4,n}\\ a_{k,5} & \cdots \, & a_{k,n}\end{array}\right)\in M_{5,n-4}(\mathbb{Z}).\end{eqnarray}$$
According to our assumption, one has
$\text{rank}(B)\leqslant 4$
. Then we conclude from above that
${\it\gamma}_{k}$
can be linear represented by
${\it\gamma}_{1}$
,
${\it\gamma}_{2}$
,
${\it\gamma}_{3}$
and
${\it\gamma}_{4}$
. Therefore, one has
$\text{rank}(H)=4$
, where
$$\begin{eqnarray}H=\left(\begin{array}{@{}ccc@{}}a_{1,5} & \cdots \, & a_{1,n}\\ \vdots & \cdots \, & \vdots \\ a_{n,5} & \cdots \, & a_{n,n}\end{array}\right)\in M_{n,n-4}(\mathbb{Z}).\end{eqnarray}$$
We obtain
$\text{rank}(A)\leqslant \text{rank}(H)+4\leqslant 8$
. This is contradictory to the condition that
$\text{rank}(A)\geqslant 9$
. Therefore, we complete the proof of Theorem 1.1.
