Open Access

One Diophantine inequality with integer and prime variables

Journal of Inequalities and Applications20152015:293

DOI: 10.1186/s13660-015-0817-y

Received: 26 June 2015

Accepted: 9 September 2015

Published: 18 September 2015

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.

Keywords

Davenport-Heilbronn method prime Diophantine approximation

MSC

11D75 11P55

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}$$
 □

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

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11371122, 11471112), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (China).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Computer and Information Engineering, Henan University of Economics and Law
(2)
Department of Mathematics and Information Science, Henan University of Economics and Law

References

  1. Vaughan, RC: The Hardy-Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997) MATHView ArticleGoogle Scholar
  2. Vaughan, RC: Diophantine approximation by prime numbers, I. Proc. Lond. Math. Soc. 28, 373-384 (1974) MATHMathSciNetView ArticleGoogle Scholar
  3. Vaughan, RC: Diophantine approximation by prime numbers, II. Proc. Lond. Math. Soc. 28, 385-401 (1974) MATHMathSciNetView ArticleGoogle Scholar
  4. Davenport, H, Roth, KF: The solubility of certain Diophantine inequalities. Mathematika 2, 81-96 (1955) MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Yang and Li 2015