One Diophantine inequality with integer and prime variables

Abstract

In this paper, we show that if $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$ are positive real numbers, at least one of the ratios $\lambda_{i}/\lambda_{j}$ ($1\leq i< j\leq4$) is irrational, then the inequality $|\lambda_{1}x_{1}^{2}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4} +\lambda_{4}x_{4}^{5}-p-\frac{1}{2}|<\frac{1}{2}$ has infinite solutions with natural numbers $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ and prime p.

1 Introduction

Diophantine inequalities with integer or prime variables have been considered by many scholars. The present paper investigates one diophantine inequality with integer and prime variables. Using the Davenport-Heilbronn method, we establish our result as follows.

Theorem 1.1

Let $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$ be positive real numbers, at least one of the ratios $\lambda_{i}/\lambda_{j}$ ($1\leq i< j\leq4$) is irrational. Then the inequality

$$\biggl|\lambda_{1}x_{1}^{2}+\lambda_{2}x_{2}^{3}+ \lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{5}-p- \frac {1}{2}\biggr|< \frac{1}{2} $$

has infinite solutions with natural numbers $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ and prime p.

2 Notation and outline of the proof

Throughout, we use p to denote a prime number and $x_{j}$ to denote a natural number. We denote by δ a sufficiently small positive number and by ε an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$. We write $e(x)=\exp(2\pi i x)$. We use $[x]$ to denote the integer part of real variable x. We take X to be the basic parameter, a large real integer. Since at least one of the ratios $\lambda_{i}/\lambda_{j}$ ($1\leq i< j\leq4$) is irrational, without loss of generality we may assume that $\lambda_{1}/ \lambda_{2}$ is irrational. For the other cases, the only difference is in the following intermediate region, and we may deal with the same method in Section 4.

Since $\lambda_{1}/ \lambda_{2}$ is irrational, then there are infinitely many pairs of integers q, a with $|\lambda_{1}/\lambda _{2}-a/q|\leq q^{-2}$, $(a,q)=1$, $q>0$ and $a\neq 0$. We choose q to be large in terms of $\lambda_{1}$, $\lambda_{2}$, $\lambda _{3}$, $\lambda_{4}$ and make the following definitions:

$$\begin{aligned}& N\asymp X^{2},\qquad L=\log N,\qquad \bigl[N^{1-8\delta} \bigr]=q, \qquad \tau=N^{-1+\delta},\\& Q= \bigl(|\lambda_{1}|^{-1}+| \lambda_{2}|^{-1} \bigr)N^{1-\delta},\qquad P=N^{6\delta},\qquad T=N^{\frac{1}{3}}. \end{aligned}$$

Let ν be a positive real number, we define

$$ \begin{aligned} &K_{\nu}(\alpha)=\nu \biggl( \frac{\sin\pi \nu\alpha}{\pi\nu\alpha} \biggr)^{2},\quad\alpha\neq0,\qquad K_{\nu}(0)=\nu, \\ &F_{1}(\alpha)=\sum_{1\leq x\leq X}e \bigl(\alpha x^{2} \bigr),\qquad F_{2}(\alpha)=\sum _{1\leq x\leq X^{\frac{2}{3}}}e \bigl(\alpha x^{3} \bigr),\qquad F_{3}(\alpha)=\sum_{1\leq x\leq X^{\frac{1}{2}}}e \bigl(\alpha x^{4} \bigr), \\ &F_{4}(\alpha)=\sum_{1\leq x\leq X^{\frac{2}{5}}}e \bigl(\alpha x^{5} \bigr),\qquad G(\alpha)=\sum_{p\leq N}( \log p)e(\alpha p), \\ &f_{1}(\alpha)=\int_{1}^{X}e \bigl( \alpha x^{2} \bigr)\,dx,\qquad f_{2}(\alpha)=\int _{1}^{X^{\frac{2}{3}}}e \bigl(\alpha x^{3} \bigr)\,dx, \qquad f_{3}(\alpha)=\int_{1}^{X^{\frac{1}{2}}}e \bigl( \alpha x^{4} \bigr)\,dx, \\ &f_{4}(\alpha)=\int_{1}^{X^{\frac{2}{5}}}e \bigl( \alpha x^{5} \bigr)\,dx,\qquad g(\alpha)=\int_{1}^{N}e( \alpha x)\,dx. \end{aligned} $$

(2.1)

It follows from (2.1) that

$$\begin{aligned}& K_{\nu}(\alpha)\ll\min \bigl(\nu,\nu^{-1}| \alpha|^{-2} \bigr), \end{aligned}$$

(2.2)

$$\begin{aligned}& \int_{-\infty}^{+\infty}e(\alpha y)K_{\nu}( \alpha)\,d\alpha=\max \bigl(0,1-\nu^{-1}|y| \bigr). \end{aligned}$$

(2.3)

From (2.3) it is clear that

$$\begin{aligned} J =:& \int_{-\infty}^{+\infty}\prod _{i=1}^{4}F_{i}(\lambda_{i} \alpha) G(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}( \alpha)\,d\alpha \\ \leq& \log N\mathop{\sum_{|\lambda_{1}x_{1}^{2}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{5} -p-\frac{1}{2}|< \frac{1}{2}}}_{1\leq x_{1}\leq X,1\leq x_{2}\leq X^{2/3}, 1\leq x_{3}\leq X^{1/2},1\leq x_{4} \leq X^{2/5}, p\leq N}1 \\ =:& (\log N){\mathcal{N}}(X), \end{aligned}$$

thus

$${\mathcal{N}}(X)\geq(\log N)^{-1}J. $$

To estimate J, we split the range of infinite integration into three sections, traditional named the neighborhood of the origin $\frak{C}=\{\alpha\in{\mathbb{R}}:|\alpha|\leq\tau\}$, the intermediate region $\frak{D}=\{\alpha\in{\mathbb{R}}:\tau<|\alpha |\leq P\}$, the trivial region $\frak{c}=\{\alpha\in{\mathbb{R}}:|\alpha|>P\}$.

To prove Theorem 1.1, we shall establish that

$$J({\frak{C}})\gg X^{\frac{77}{30}},\qquad J({\frak{D}})=o \bigl(X^{\frac{70}{33}} \bigr),\qquad J({\frak{c}})=o \bigl(X^{\frac{77}{30}} \bigr) $$

in Sections 3, 4 and 5, respectively. Thus

$$J\gg X^{\frac{77}{30}},\qquad {\mathcal{N}}(X)\gg X^{\frac{77}{30}}L^{-1}, $$

and Theorem 1.1 can be established.

3 The neighborhood of the origin

Lemma 3.1

If $\alpha=a/q+\beta$, where $(a,q)=1$, then

$$\sum_{1\leq x\leq N^{1/t}}e \bigl(\alpha x^{t} \bigr)=q^{-1}\sum_{m=1}^{q}e \bigl(am^{t}/q \bigr)\int_{1}^{N^{1/t}}e \bigl( \beta y^{t} \bigr)\,dy+O \bigl(q^{1/2+\varepsilon }\bigl(1+N|\beta|\bigr) \bigr). $$

Proof

This is Theorem 4.1 of Vaughan [1]. □

If $|\alpha|\in\frak{C}$, by Lemma 3.1, taking $a=0$, $q=1$, then

$$ F_{i}(\alpha)=f_{i}(\alpha)+O \bigl(X^{2\delta} \bigr), \quad i=1,2,3,4. $$

(3.1)

Lemma 3.2

Let $\rho=\beta+i\gamma$ be a typical zero of the Riemann zeta function, C be a positive constant,

$$I(\alpha)=\sum_{|\gamma|\leq T, \beta\geq \frac{2}{3}}\sum _{n\leq N}n^{\rho-1}e(n\alpha),\qquad J(\alpha)=O \bigl(\bigl(1+| \alpha|N\bigr)N^{\frac{2}{3}}L^{C} \bigr), $$

then

$$\begin{aligned}& G(\alpha)=g(\alpha)-I(\alpha)+J(\alpha), \end{aligned}$$

(3.2)

$$\begin{aligned}& \int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|I(\alpha)\bigr|^{2}\,d\alpha \ll N\exp \bigl(-L^{\frac{1}{5}} \bigr), \end{aligned}$$

(3.3)

$$\begin{aligned}& \int_{-\tau}^{\tau}\bigl|J(\alpha)\bigr|^{2}\,d\alpha \ll N\exp \bigl(-L^{\frac{1}{5}} \bigr). \end{aligned}$$

(3.4)

Proof

Equations (3.2), (3.3), (3.4) can be seen from Lemma 5, (29) and (33) given by Vaughan [2]. □

Lemma 3.3

We have

$$\begin{aligned}& \int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|f_{1}( \alpha)\bigr|^{2}\,d\alpha \ll L^{2},\qquad \int _{-\frac{1}{2}}^{\frac{1}{2}}\bigl|f_{2}( \alpha)\bigr|^{2} \,d\alpha \ll X^{-\frac{2}{3}}L^{2}. \\& \int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|f_{3}( \alpha)\bigr|^{2}\,d\alpha \ll X^{-1} L^{2},\qquad \int _{-\frac{1}{2}}^{\frac{1}{2}}\bigl|f_{4}(\alpha)\bigr|^{2}\,d \alpha \ll X^{-\frac{6}{5}}L^{2}. \end{aligned}$$

Proof

These results are from (5.16) of Vaughan [3]. □

Lemma 3.4

We have

$$\int_{{\frak{C}}}\Biggl|\prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha) G(-\alpha)-\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) g(-\alpha)\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha\ll X^{\frac{77}{30}}L^{-1}. $$

Proof

It is obvious that $F_{1}(\lambda_{1}\alpha)\ll X$, $f_{1}(\lambda_{1}\alpha)\ll X$, $F_{2}(\lambda_{2}\alpha)\ll X^{\frac{2}{3}}$, $f_{2}(\lambda_{1}\alpha)\ll X^{\frac{2}{3}}$, $F_{3}(\lambda_{3}\alpha)\ll X^{\frac{1}{2}}$, $f_{3}(\lambda_{3}\alpha)\ll X^{\frac{1}{2}}$, $F_{4}(\lambda_{4}\alpha)\ll X^{\frac{2}{5}}$, $f_{4}(\lambda_{4}\alpha)\ll X^{\frac{2}{5}}$, $G(-\alpha)\ll N$, $g(-\alpha)\ll N$,

$$\begin{aligned} & \prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha)G(-\alpha)-\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha)g(-\alpha) \\ &\quad= \bigl(F_{1}(\lambda_{1}\alpha)-f_{1}( \lambda_{1}\alpha) \bigr)\prod_{i=2}^{4}F_{i}( \lambda_{i}\alpha)G(-\alpha) + \bigl(F_{2}( \lambda_{2}\alpha)-f_{2}(\lambda_{2}\alpha) \bigr) \mathop{\prod_{i=1}}_{i\neq2}^{4}F_{i}( \lambda_{i}\alpha)G(-\alpha) \\ &\qquad{}+ \bigl(F_{3}(\lambda_{3}\alpha)-f_{3}( \lambda_{3}\alpha) \bigr)\mathop{\prod_{i=1}}_{{i\neq3}}^{4} F_{i}(\lambda_{i}\alpha)G(-\alpha) + \bigl(F_{4}( \lambda_{4}\alpha)-f_{4}(\lambda_{4}\alpha) \bigr) \prod_{i=1}^{3}f_{i}( \lambda_{i}\alpha)G(-\alpha) \\ &\qquad{}+\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) \bigl(G(-\alpha)-g(-\alpha) \bigr). \end{aligned}$$

Then by (3.1), Lemmas 3.2 and 3.3,

$$\begin{aligned} &\int_{{\frak{C}}}\Biggl| \bigl(F_{1}(\lambda_{1} \alpha)-f_{1}(\lambda_{1}\alpha) \bigr)\prod _{i=2}^{4} F_{i}(\lambda_{i} \alpha)G(-\alpha)\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha \ll N^{-1+\delta}X^{2\delta}X^{\frac{2}{3}+\frac{1}{2}+\frac{2}{5}}N \ll X^{\frac{47}{30}+4\delta}, \\ &\int_{{\frak{C}}}\Biggl|\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) \bigl(G(-\alpha )-g(-\alpha) \bigr)\Biggr|K_{\frac{1}{2}}( \alpha)\,d\alpha \\ &\quad\ll X^{\frac{47}{30}} \biggl(\int_{{\frak{C}}}\bigl|f_{1}( \lambda_{1}\alpha )\bigr|^{2}K_{\frac{1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{2}} \biggl(\int_{{\frak{C}}}\bigl|J(-\alpha)-I(- \alpha)\bigr|^{2}K_{\frac{1}{2}}(\alpha )\,d\alpha \biggr)^{\frac{1}{2}} \\ &\quad\ll X^{\frac{47}{30}} \biggl(\int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|f_{1}( \lambda _{1}\alpha)\bigr|^{2}\,d\alpha \biggr)^{\frac{1}{2}} \biggl(\int_{{\frak{C}}}\bigl|J(\alpha)\bigr|^{2}\,d\alpha+\int _{-\frac{1}{2}}^{\frac {1}{2}}\bigl|I(\alpha)\bigr|^{2}\,d\alpha \biggr)^{\frac{1}{2}} \\ &\quad\ll X^{\frac{47}{30}}L \bigl(N\exp \bigl(-L^{\frac{1}{5}} \bigr) \bigr)^{\frac{1}{2}} \\ &\quad\ll X^{\frac{77}{30}}L^{-1}. \end{aligned}$$

The other cases are similar, and the proof of Lemma 3.4 is completed. □

Lemma 3.5

We have

$$\int_{|\alpha|>N^{-1+\delta}}\Biggl|\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) g(-\alpha)\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha\ll X^{\frac{77}{30}-\frac{77}{30}\delta}. $$

Proof

It follows from Vaughan [1] that for $\alpha\neq0$,

$$\begin{aligned}& f_{1}(\lambda_{1}\alpha)\ll|\alpha|^{-\frac{1}{2}},\qquad f_{2}(\lambda_{2}\alpha )\ll|\alpha|^{-\frac{1}{3}},\qquad f_{3}(\lambda_{3}\alpha)\ll|\alpha|^{-\frac{1}{4}}, \\& f_{4}(\lambda_{4}\alpha )\ll|\alpha|^{-\frac{1}{5}},\qquad g(-\alpha)\ll|\alpha|^{-1}. \end{aligned}$$

Thus

$$\int_{|\alpha|>N^{-1+\delta}}\Biggl|\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha )g(-\alpha)\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha \ll \int_{|\alpha|>N^{-1+\delta}}|\alpha|^{-\frac{137}{60}}\,d\alpha \ll X^{\frac{77}{30}-\frac{77}{30}\delta}. $$

 □

Lemma 3.6

We have

$$\int_{-\infty}^{+\infty}\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) g(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}(\alpha)\,d\alpha\gg X^{\frac{77}{30}}. $$

Proof

From (2.3), one has

$$\begin{aligned} & \int_{-\infty}^{+\infty}\prod _{i=1}^{4}f_{i}(\lambda_{i} \alpha) g(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}( \alpha)\,d\alpha \\ &\quad= \int_{1}^{X}\int_{1}^{X^{\frac{2}{3}}} \int_{1}^{X^{\frac{1}{2}}}\int_{1}^{X^{\frac{2}{5}}} \int_{1}^{N}\int_{-\infty}^{+\infty}e \Biggl(\alpha \Biggl(\sum_{i=1}^{4}\lambda _{i}x^{1+i}_{i}-x-\frac{1}{2} \Biggr) \Biggr) \\ &\qquad{}\cdot K_{\frac{1}{2}}(\alpha)\,d\alpha \,dx \,dx_{4}\cdots \,dx_{1} \\ &\quad= \frac{1}{120}\int_{1}^{X^{2}}\cdots\int _{1}^{X^{2}} \int_{1}^{N} \int_{-\infty}^{+\infty}x_{1}^{-\frac{1}{2}}x_{2}^{-\frac {2}{3}}x_{3}^{-\frac{3}{4}} x_{4}^{-\frac{4}{5}}e \Biggl(\alpha \Biggl(\sum _{i=1}^{4}\lambda_{i} x_{i}-x- \frac {1}{2} \Biggr) \Biggr) \\ & \qquad{} \cdot K_{\frac{1}{2}}(\alpha)\,d\alpha \,dx \,dx_{4}\cdots \,dx_{1} \\ &\quad= \frac{1}{120}\int_{1}^{X^{2}}\cdots\int _{1}^{X^{2}}\int_{1}^{N}x_{1}^{-\frac {1}{2}}x_{2}^{-\frac{2}{3}}x_{3}^{-\frac{3}{4}} x_{4}^{-\frac{4}{5}} \\ &\qquad{}\cdot\max \Biggl(0,\frac{1}{2}-\Biggl|\sum _{i=1}^{4}\lambda_{i} x_{i}-x- \frac{1}{2}\Biggr| \Biggr)\,dx \,dx_{4}\cdots \,dx_{1}. \end{aligned}$$

Let $|\sum_{i=1}^{4}\lambda_{i} x_{i}-x-\frac{1}{2}|\leq\frac{1}{4}$, then $\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{3}{4}\leq x\leq \sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{1}{4}$. Based on $\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{3}{4}>1$, $\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{1}{4}< N$, one may take

$$\lambda_{j}X^{2} \Biggl(8\sum_{i=1}^{4} \lambda_{i} \Biggr)^{-1} \leq x_{j} \leq \lambda_{j}X^{2} \Biggl(4\sum_{i=1}^{4} \lambda_{i} \Biggr)^{-1},\quad j=1,\ldots,4, $$

hence

$$\int_{-\infty}^{+\infty}\prod_{i=1}^{4}f_{i}( \lambda_{i}\alpha) g(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}(\alpha)\,d\alpha \geq\frac{1}{960}\prod _{j=1}^{4}\lambda_{j} \Biggl(8\sum _{i=1}^{4}\lambda_{i} \Biggr)^{-4}X^{\frac{77}{30}}. $$

This completes the proof of Lemma 3.6. □

4 The intermediate region

Lemma 4.1

We have

$$\begin{aligned}& \int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{4}K_{\frac{1}{2}}(\alpha )\,d\alpha \ll X^{2+\varepsilon}, \end{aligned}$$

(4.1)

$$\begin{aligned}& \int_{-\infty}^{+\infty}\bigl|F_{2}( \lambda_{2}\alpha)\bigr|^{8}K_{\frac{1}{2}}(\alpha )\,d\alpha \ll X^{\frac{10}{3}+\frac{2}{3}\varepsilon}, \end{aligned}$$

(4.2)

$$\begin{aligned}& \int_{-\infty}^{+\infty}\bigl|F_{3}( \lambda_{3}\alpha)\bigr|^{16}K_{\frac {1}{2}}(\alpha)\,d\alpha \ll X^{6+\frac{1}{2}\varepsilon}, \end{aligned}$$

(4.3)

$$\begin{aligned}& \int_{-\infty}^{+\infty}\bigl|F_{4}( \lambda_{4}\alpha)\bigr|^{32}K_{\frac {1}{2}}(\alpha)\,d\alpha \ll X^{\frac{54}{5}+\frac{2}{5}\varepsilon}, \end{aligned}$$

(4.4)

$$\begin{aligned}& \int_{-\infty}^{+\infty}\bigl|G(-\alpha)\bigr|^{2}K_{\frac{1}{2}}( \alpha)\,d\alpha \ll NL. \end{aligned}$$

(4.5)

Proof

By (2.2) and Hua’s inequality, we have

$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{4}K_{\frac{1}{2}}(\alpha )\,d\alpha \\ &\quad\ll \sum_{m=-\infty}^{+\infty}\int _{m}^{m+1}\bigl|F_{1}(\lambda_{1} \alpha )\bigr|^{4}K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \sum_{m=0}^{1}\int _{m}^{m+1}\bigl|F_{1}(\lambda_{1} \alpha)\bigr|^{4}\,d\alpha +\sum_{m=2}^{+\infty}m^{-2} \int_{m}^{m+1}\bigl|F_{1}( \lambda_{1}\alpha )\bigr|^{4}\,d\alpha \\ &\quad\ll X^{2+\varepsilon}+X^{2+\varepsilon}\sum_{m=2}^{+\infty}m^{-2} \\ &\quad\ll X^{2+\varepsilon}. \end{aligned}$$

The proofs of (4.2)-(4.5) are similar to (4.1). □

Lemma 4.2

We have

$$\int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{2}\bigl|F_{3}(\lambda_{3} \alpha )\bigr|^{4}K_{\frac{1}{2}}(\alpha)\,d\alpha \ll X^{2+\varepsilon}. $$

Proof

Firstly, we consider the number of solutions $R(X,Z)$ of equation

$$\lambda_{1} \bigl(x_{1}^{2}-x_{2}^{2} \bigr)=\lambda_{j} \bigl(y_{1}^{4}+y_{2}^{4}-y_{3}^{4}-y_{4}^{4} \bigr),\quad 1\leq x_{1},x_{2}\leq X, 1\leq y_{1},y_{2},y_{3},y_{4}\leq Z. $$

If $x_{1}=x_{2}$, then $R(X,Z)\ll X^{\varepsilon}XZ^{2}$, and if $x_{1}\neq x_{2}$, then $R(X,Z)\ll X^{\varepsilon}Z^{4}$. We take $Z=X^{\frac{1}{2}}$, then $R(X,Z)\ll X^{2+\varepsilon}$.

$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{2}\bigl|F_{3}(\lambda_{3} \alpha )\bigr|^{4}K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \sum_{m=-\infty}^{+\infty}\int _{m}^{m+1}\bigl|F_{1}(\lambda_{1} \alpha )\bigr|^{2}\bigl|F_{3}(\lambda_{3}\alpha)\bigr|^{4} K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \sum_{m=0}^{1}\int _{m}^{m+1}\bigl|F_{1}(\lambda_{1} \alpha)\bigr|^{2}\bigl|F_{3}(\lambda _{3}\alpha)\bigr|^{4}d \alpha +\sum_{m=2}^{+\infty}m^{-2}\int _{m}^{m+1}\bigl|F_{1}(\lambda_{1} \alpha )\bigr|^{2}\bigl|F_{3}(\lambda_{3}\alpha)\bigr|^{4}d \alpha \\ &\quad\ll X^{2+\varepsilon}. \end{aligned}$$

 □

Lemma 4.3

Suppose that $(a,q)=1$, $|\alpha-a/q|\leq q^{-2}$, $\phi (x)=\alpha x^{k}+\alpha_{1}x^{k-1}+\cdots+\alpha_{k-1}x+\alpha_{k}$, then

$$\sum_{x=1}^{M}e \bigl(\phi(x) \bigr)\ll M^{1+\varepsilon } \bigl(q^{-1}+M^{-1}+qM^{-k} \bigr)^{2^{1-k}}. $$

Proof

This is Lemma 2.4 (Weyl’s inequality) of Vaughan [1]. □

Lemma 4.4

For every real number $\alpha\in\frak{D}$, let $W(\alpha)=\min(|F_{1}(\lambda_{1}\alpha)|^{\frac{2}{3}},|F_{2}(\lambda _{2}\alpha)|)$, then

$$W(\alpha)\ll X^{\frac{2}{3}-\frac{1}{3}\delta+\varepsilon}. $$

Proof

For $\alpha\in\frak{D}$ and $j=1,2$, we choose $a_{j}$, $q_{j}$ such that

$$ |\lambda_{j}\alpha-a_{j}/q_{j}|\leq q_{j}^{-1}Q^{-1} $$

(4.6)

with $(a_{j},q_{j})=1$ and $1\leq q_{j}\leq Q$.

Firstly, we note that $a_{1}a_{2}\neq0$. Secondly, if $q_{1},q_{2}\leq P$, then

$$\biggl|a_{2}q_{1}\frac{\lambda_{1}}{\lambda_{2}}-a_{1}q_{2}\biggr| \leq \biggl|\frac{a_{2}/q_{2}}{\lambda_{2}\alpha}q_{1}q_{2} \biggl( \lambda_{1}\alpha-\frac{a_{1}}{q_{1}} \biggr)\biggr|+ \biggl|\frac{a_{1}/q_{1}}{\lambda_{2}\alpha}q_{1}q_{2} \biggl(\lambda_{2}\alpha-\frac{a_{2}}{q_{2}} \biggr)\biggr| \ll PQ^{-1}< \frac{1}{2q}. $$

We recall that q was chosen as the denominator of a convergent to the continued fraction for $\lambda_{1}/\lambda_{2}$. Thus, by Legendre’s law of best approximation, we have $|q'\frac{\lambda_{1}}{\lambda_{2}}-a'|>\frac{1}{2q}$ for all integers $a'$, $q'$ with $1\leq q'< q$, thus $|a_{2}q_{1}|\geq q=[N^{1-8\delta}]$. However, from (4.6) we have $|a_{2}q_{1}|\ll q_{1}q_{2}P \ll N^{18\delta}$, this is a contradiction. We have thus established that for at least one j, $P< q_{j}\ll Q$. Hence, Lemma 4.3 gives the desired inequality for $W(\alpha)$. □

Lemma 4.5

We have

$$\int_{\frak{D}}\prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha) G(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}(\alpha)\,d\alpha \ll X^{\frac{77}{30}-\frac{1}{12}\delta+\varepsilon}. $$

Proof

By Lemmas 4.1, 4.2, 4.4 and Hölder’s inequality, we have

$$\begin{aligned} & \int_{{\frak{D}}}\prod_{i=1}^{4}\bigl|F_{i}( \lambda_{i}\alpha)G(-\alpha )\bigr|K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \max_{\alpha\in{\frak{D}}}\bigl|W(\alpha)\bigr|^{\frac{3}{16}} \int _{{\frak{D}}}\bigl|F_{1}(\lambda_{1} \alpha)\bigr|^{\frac{7}{8}}\prod_{i=2}^{4} \bigl|F_{i}(\lambda_{i}\alpha)G(-\alpha)\bigr|K_{\frac{1}{2}}( \alpha)\,d\alpha \\ & \qquad{} +\max_{\alpha\in{\frak{D}}}\bigl|W(\alpha)\bigr|^{\frac{1}{4}} \int _{{\frak{D}}}\bigl|F_{2}(\lambda_{2} \alpha)\bigr|^{\frac{3}{4}}\mathop{\prod_{i=1}}_{i\neq2}^{4} \bigl|F_{i}(\lambda_{i}\alpha)G(-\alpha)\bigr|K_{\frac{1}{2}}( \alpha)\,d\alpha \\ &\quad\ll \bigl(X^{\frac{2}{3}-\frac{1}{3}\delta+\varepsilon} \bigr)^{\frac{3}{16}} \biggl(\int _{-\infty}^{+\infty}\bigl|F_{1}(\lambda_{1} \alpha)\bigr|^{4}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{3}{32}} \biggl(\int_{-\infty}^{+\infty}\bigl|F_{2}( \lambda_{2}\alpha)\bigr|^{8}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{8}} \\ & \qquad{} \cdot \biggl(\int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{2}\bigl|F_{3}(\lambda _{3}\alpha)\bigr|^{4} K_{\frac{1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{4}} \biggl(\int_{-\infty}^{+\infty}\bigl|F_{4}( \lambda_{4}\alpha)\bigr|^{32}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{32}} \\ & \qquad{} \cdot \biggl(\int_{-\infty}^{+\infty}\bigl|G(- \alpha)\bigr|^{2}K_{\frac{1}{2}}(\alpha )\,d\alpha \biggr)^{\frac{1}{2}} \\ & \qquad{} + \bigl(X^{\frac{2}{3}-\frac{1}{3}\delta+\varepsilon} \bigr)^{\frac{1}{4}} \biggl(\int _{-\infty}^{+\infty}\bigl|F_{1}(\lambda_{1} \alpha)\bigr|^{4}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{8}} \biggl(\int_{-\infty}^{+\infty}\bigl|F_{2}( \lambda_{2}\alpha)\bigr|^{8}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{3}{32}} \\ & \qquad{} \cdot \biggl(\int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{2}\bigl|F_{3}(\lambda _{3}\alpha)\bigr|^{4} K_{\frac{1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{4}} \biggl(\int_{-\infty}^{+\infty}\bigl|F_{4}( \lambda_{i}\alpha)\bigr|^{32}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{32}} \\ & \qquad{}\cdot \biggl(\int_{-\infty}^{+\infty}\bigl|G(- \alpha)\bigr|^{2}K_{\frac{1}{2}}(\alpha )\,d\alpha \biggr)^{\frac{1}{2}} \\ &\quad\ll \bigl(X^{\frac{2}{3}-\frac{1}{3}\delta+\varepsilon} \bigr)^{\frac {3}{16}} \bigl(X^{2+\varepsilon} \bigr)^{\frac{3}{32}} \bigl(X^{\frac{10}{3}+\frac{2}{3}\varepsilon} \bigr)^{\frac{1}{8}} \bigl(X^{2+\varepsilon } \bigr)^{\frac{1}{4}} \bigl(X^{\frac{54}{5}+\frac{2}{5}\varepsilon} \bigr)^{\frac{1}{32}}(N L)^{\frac {1}{2}} \\ & \qquad{} + \bigl(X^{\frac{2}{3}-\frac{1}{3}\delta+\varepsilon} \bigr)^{\frac {1}{4}} \bigl(X^{2+\varepsilon} \bigr)^{\frac{1}{8}} \bigl(X^{\frac{10}{3}+\frac{2}{3}\varepsilon} \bigr)^{\frac {3}{32}} \bigl(X^{2+\varepsilon} \bigr)^{\frac{1}{4}} \bigl(X^{\frac{54}{5}+\frac{2}{5}\varepsilon} \bigr)^{\frac{1}{32}}(N L)^{\frac {1}{2}} \\ &\quad\ll X^{\frac{77}{30}-\frac{1}{12}\delta+\varepsilon}. \end{aligned}$$

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5 The trivial region

Lemma 5.1

(Lemma 2 of [4])

Let $V(\alpha)=\sum e(\alpha f(x_{1},\ldots,x_{m}))$, where f is any real function and the summation is over any finite set of values of $x_{1},\ldots,x_{m}$. Then, for any $A>4$, we have

$$\int_{|\alpha|>A}\bigl|V(\alpha)\bigr|^{2}K_{\nu}(\alpha) \,d\alpha \leq\frac{16}{A}\int_{-\infty}^{\infty}\bigl|V( \alpha)\bigr|^{2} K_{\nu}(\alpha)\,d\alpha. $$

Lemma 5.2

We have

$$\int_{\frak{c}}\prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha) G(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}(\alpha)\,d\alpha \ll X^{\frac{77}{30}-12\delta+\varepsilon}. $$

Proof

By Lemmas 5.1, 4.1, 4.2 and Schwarz’s inequality, we have

$$\begin{aligned} & \int_{\frak{c}}\prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha) G(-\alpha)e \biggl(-\frac{1}{2}\alpha \biggr)K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \int_{\frak{c}}\Biggl|\prod_{i=1}^{4}F_{i}( \lambda_{i}\alpha)G(-\alpha )\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll \frac{1}{P}\int_{-\infty}^{+\infty}\Biggl|\prod _{i=1}^{4}F_{i}(\lambda _{i}\alpha) G(-\alpha)\Biggr|K_{\frac{1}{2}}(\alpha)\,d\alpha \\ &\quad\ll N^{-6\delta}\max\bigl|F_{4}(\lambda_{4}\alpha)\bigr| \biggl(\int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{4}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{8}} \\ & \qquad{} \cdot \biggl(\int_{-\infty}^{+\infty}\bigl|F_{1}( \lambda_{1}\alpha)\bigr|^{2}\bigl|F_{3}(\lambda _{3}\alpha)\bigr|^{4} K_{\frac{1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{4}} \biggl(\int_{-\infty}^{+\infty}\bigl|F_{2}( \lambda_{2}\alpha)\bigr|^{8}K_{\frac {1}{2}}(\alpha)\,d\alpha \biggr)^{\frac{1}{8}} \\ & \qquad{} \cdot \biggl(\int_{-\infty}^{+\infty}\bigl|G(- \alpha)\bigr|^{2}K_{\frac{1}{2}}(\alpha )\,d\alpha \biggr)^{\frac{1}{2}} \\ &\quad\ll N^{-6\delta}X^{\frac{2}{5}} \bigl(X^{2+\varepsilon} \bigr)^{\frac{1}{8}+\frac{1}{4}} \bigl(X^{\frac{10}{3}+\frac{2}{3}\varepsilon} \bigr)^{\frac{1}{8}}(N L)^{\frac {1}{2}} \\ &\quad\ll X^{\frac{77}{30}-12\delta+\varepsilon}. \end{aligned}$$

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