Open Access

One Diophantine inequality with unlike powers of prime variables

Journal of Inequalities and Applications20162016:33

https://doi.org/10.1186/s13660-016-0983-6

Received: 18 November 2015

Accepted: 19 January 2016

Published: 28 January 2016

Abstract

In this paper, we show that if \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda _{4}\), \(\lambda_{5}\) are nonzero real numbers not all of the same sign, η is real, \(0<\sigma<\frac{1}{720}\), and at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq5\)) is irrational, then the inequality \(|\lambda_{1}p_{1}+\lambda_{2}p_{2}^{2}+\lambda_{3}p_{3}^{3}+\lambda_{4}p_{4}^{4}+\lambda _{5}p_{5}^{5}+\eta|<(\max_{ 1\leq j\leq5}{p_{j}^{j}})^{-\sigma}\) has infinite solutions with primes \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\), \(p_{5}\).

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. Recently, Yang and Li in [1] proved that the inequality
$$\begin{aligned} \biggl|\lambda_{1}x_{1}^{2}+\lambda_{2}x_{2}^{3}+ \lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{5}-p- \frac {1}{2} \biggr|< \frac{1}{2} \end{aligned}$$
has infinite solutions with natural numbers \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\) and prime p. Using the Davenport-Heilbronn method, we establish our result as follows.

Theorem 1.1

Let \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\) be nonzero real numbers not all of the same sign, η is real, \(0<\sigma<\frac {1}{720}\), and at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq5\)) is irrational, then the inequality
$$\bigl|\lambda_{1}p_{1}+\lambda_{2}p_{2}^{2}+ \lambda_{3}p_{3}^{3}+\lambda_{4}p_{4}^{4}+ \lambda _{5}p_{5}^{5}+\eta\bigr|< \Bigl(\max _{ 1\leq j\leq5}{p_{j}^{j}}\Bigr)^{-\sigma} $$
has infinite solutions with primes \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\), \(p_{5}\).

2 Notation and outline of the proof

Throughout, we use p to denote a prime number. We denote by δ a sufficiently small positive number and by ε an arbitrarily small positive number, not necessarily the same at different occurrences. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\), and η. We write \(e(x)=e^{2\pi ix}\). We take X to be the basic parameter, a large real integer. Since at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq5\)) 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, 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 \(a\neq0\). We choose q to be large in terms of \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda _{3}\), \(\lambda_{4}\), \(\lambda_{5}\), η and make the following definitions:
$$\begin{aligned}& N=q^{2}, \qquad L=\log N,\qquad 0< \sigma< \frac{\theta}{32}< \frac{1}{720}, \nu =N^{-\sigma}, \qquad \tau=N^{-1+\theta}, \end{aligned}$$
(2.1)
$$\begin{aligned}& P=N^{\theta}L^{-1},\qquad Q=\bigl(|\lambda_{1}|^{-1}+| \lambda_{2}|^{-1}\bigr)N^{1-\theta},\qquad T_{1}=T_{2}^{2}=T_{3}^{3}=T_{4}^{4}=T_{5}^{5}=N^{\frac{1}{3}}. \end{aligned}$$
(2.2)
Let u be a positive real number, we define
$$\begin{aligned}& K_{u}(\alpha)= \biggl(\frac{\sin{\pi u\alpha}}{\pi\alpha} \biggr)^{2}\quad(\alpha \neq0),\qquad K_{u}(0)=u^{2}, \end{aligned}$$
(2.3)
$$\begin{aligned}& F_{k}(\alpha)=\sum_{(\delta N)^{1/k}\leq p\leq N^{1/k}}e\bigl(\lambda _{k}p^{k}\alpha\bigr)\log p,\quad k=1,2,3,4,5, \end{aligned}$$
(2.4)
$$\begin{aligned}& I_{k}(\alpha)= \int_{(\delta N)^{1/k}}^{N^{1/k}}e\bigl(\lambda_{k}y^{k} \alpha\bigr) \,\mathrm{d}y,\quad k=1,2,3,4,5, \end{aligned}$$
(2.5)
$$\begin{aligned}& J_{k}(\alpha)=\mathop{\sum_{|\gamma|\leq T_{k} }}_{\beta\geq\frac{2}{3}}\sum _{\delta N< n\leq N}n^{-1+\rho/k}e(\lambda_{k}\alpha n), \quad k=1,2,3,4,5, \end{aligned}$$
(2.6)
where \(\rho=\beta+i\gamma(\beta,\gamma \mbox{ real})\) is a typical non-trivial zero of the Riemann Zeta function.
It follows from (2.3) that
$$ K_{u}(\alpha)\ll\min\bigl(u^{2},| \alpha|^{-2}\bigr),\qquad \int_{-\infty}^{+\infty }e(\alpha y)K_{u}(\alpha) \,\mathrm{d}\alpha=\max\bigl(0,u-|y|\bigr). $$
(2.7)
From (2.7) it is clear that
$$\begin{aligned} J&:= \int_{-\infty}^{+\infty}\prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta)K_{\nu }(\alpha)\,\mathrm{d} \alpha \\ &\leq(\log N)^{5}\mathop{\sum_{|\lambda_{1}p_{1}+\lambda_{2}p_{2}^{2}+\lambda _{3}p_{3}^{3}+\lambda_{4}p_{4}^{4}+\lambda_{5}p_{5}^{5}+\eta|< \nu}}_{(\delta N)^{1/k} \leq p_{k}\leq N^{1/k}, k=1,2,3,4,5}1 \\ &=:(\log N)^{5}\mathcal{N}(N). \end{aligned}$$
Thus we have
$$\mathcal{N}(N)\geq(\log N)^{-5}J. $$

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

To prove Theorem 1.1, we shall establish that
$$J(\mathfrak{C})\gg\nu^{2}N^{\frac{77}{60}}, \qquad J(\mathfrak{D})=o \bigl(\nu ^{2}N^{\frac{77}{60}}\bigr),\qquad J(\mathfrak{c})=o\bigl( \nu^{2}N^{\frac{77}{60}}\bigr) $$
in Sections 3, 4, and 5, respectively. Thus
$$\mathcal{N}(N)\gg\nu^{2}(\log N)^{-5}N^{\frac{77}{60}}, $$
and Theorem 1.1 can be established.

3 The neighborhood of the origin

We let
$$\begin{aligned} B_{k}(\alpha)=F_{k}(\alpha)-I_{k}( \alpha)+J_{k}(\alpha),\quad k=1,2,3,4,5. \end{aligned}$$
(3.1)
We use C to denote a positive absolute constant, not necessarily the same one on each occurrence.

Lemma 3.1

We have
$$\begin{aligned} B_{k}(\alpha)\ll N^{\frac{2}{3k}}L^{C}\bigl(1+| \alpha|N\bigr), \quad k=1,2,3,4,5. \end{aligned}$$
(3.2)
This is Lemma 7 of Vaughan [2].

Lemma 3.2

For \(k=1,2,3,4,5\), we have
$$\begin{aligned} &I_{k}(\alpha)\ll N^{\frac{1}{k}}\min \bigl(1,N^{-1}|\alpha|^{-1}\bigr), \end{aligned}$$
(3.3)
$$\begin{aligned} & \int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|J_{k}(\alpha)\bigr|^{2} \,\mathrm{d}\alpha\ll N^{\frac{2}{k}-1}\exp\bigl(-2L^{-\frac{1}{5}}\bigr), \end{aligned}$$
(3.4)
$$\begin{aligned} & \int_{-\frac{1}{2}}^{\frac{1}{2}}\bigl|I_{k}(\alpha)\bigr|^{2} \,\mathrm{d}\alpha\ll N^{\frac{2}{k}-1}, \end{aligned}$$
(3.5)
$$\begin{aligned} & \int_{-\tau}^{\tau}\bigl|B_{k}(\alpha)\bigr|^{2} \,\mathrm{d}\alpha\ll N^{\frac {2}{k}-1}\exp\bigl(-2L^{-\frac{1}{5}}\bigr), \end{aligned}$$
(3.6)
$$\begin{aligned} & \int_{-\tau}^{\tau}\bigl|F_{k}(\alpha)\bigr|^{2} \,\mathrm{d}\alpha\ll N^{\frac{2}{k}-1}. \end{aligned}$$
(3.7)

Proof

The inequality (3.6) follows from (2.1) and Lemma 3.1. The others are similar to Lemma 8 of Vaughan [2]. □

Lemma 3.3

We have
$$\begin{aligned} \int_{\mathfrak{C}}\Biggl\vert \prod_{i=1}^{5}F_{i}( \alpha)-\prod_{i=1}^{5}I_{i}( \alpha)\Biggr\vert K_{\nu}(\alpha) \,\mathrm{d}\alpha\ll \nu^{2}N^{\frac{77}{60}}\exp\bigl(-L^{-\frac{1}{5}}\bigr). \end{aligned}$$
(3.8)

Proof

Note that
$$\begin{aligned} &\prod_{i=1}^{5}F_{i}(\alpha)- \prod_{i=1}^{5}I_{i}(\alpha) \\ &\quad=\bigl(F_{1}(\alpha)-I_{1}(\alpha)\bigr)\prod _{i=2}^{5}F_{i}(\alpha) +I_{1}(\alpha) \bigl(F_{2}(\alpha)-I_{2}(\alpha) \bigr)\prod_{i=3}^{5}F_{i}(\alpha) \\ &\qquad{}+I_{1}(\alpha)I_{2}(\alpha) \bigl(F_{3}( \alpha)-I_{3}(\alpha)\bigr)F_{4}(\alpha)F_{5}( \alpha) +\prod_{i=1}^{3}I_{i}( \alpha) \bigl(F_{4}(\alpha)-I_{4}(\alpha)\bigr)F_{5}( \alpha) \\ &\qquad{}+\prod_{i=1}^{4}I_{i}( \alpha) \bigl(F_{5}(\alpha)-I_{5}(\alpha)\bigr). \end{aligned}$$
Then by (2.7), (3.1), Lemma 3.2,
$$\begin{aligned} & \int_{\mathfrak{C}}\Biggl\vert \bigl(F_{1}( \alpha)-I_{1}(\alpha)\bigr)\prod_{i=2}^{5}F_{i}( \alpha)\Biggr\vert K_{\nu}(\alpha)\,\mathrm{d}\alpha \\ &\quad\ll\nu^{2}N^{\frac{47}{60}} \int_{-\tau}^{\tau}\bigl|\bigl(B_{1}( \alpha)-J_{1}(\alpha )\bigr)F_{2}(\alpha)\bigr|\,\mathrm{d}\alpha \\ &\quad\ll\nu^{2}N^{\frac{47}{60}} \biggl( \int_{-\tau}^{\tau}\bigl|\bigl(B_{1}( \alpha)-J_{1}(\alpha)\bigr)\bigr|^{2}\,\mathrm{d}\alpha \biggr)^{\frac{1}{2}} \biggl( \int_{-\tau}^{\tau}\bigl|F_{2}(\alpha)\bigr|^{2} \,\mathrm{d}\alpha \biggr)^{\frac {1}{2}} \\ &\quad\ll\nu^{2}N^{\frac{47}{60}} \biggl( \int_{-\tau}^{\tau}\bigl(\bigl|\bigl(B_{1}( \alpha)\bigr|^{2}+\bigl|J_{1}(\alpha)\bigr)\bigr|^{2}\bigr)\,\mathrm {d}\alpha \biggr)^{\frac{1}{2}} \\ &\quad\ll\nu^{2}N^{\frac{77}{60}}\exp\bigl(-L^{-\frac{1}{5}}\bigr). \end{aligned}$$
The other cases are similar, and the proof of Lemma 3.3 is completed. □

Lemma 3.4

We have
$$\begin{aligned} \int_{|\alpha|>\tau}\Biggl\vert \prod_{i=1}^{5}I_{i}( \alpha)\Biggr\vert K_{\nu}(\alpha )\,\mathrm{d}\alpha \ll \nu^{2}N^{\frac{77}{60}-4\theta}. \end{aligned}$$
(3.9)

It follows from (2.7) and (3.3).

Lemma 3.5

We have
$$\begin{aligned} \int_{-\infty}^{+\infty}\prod_{j=1}^{5}I_{j}( \alpha)e(\alpha\eta)K_{\nu }(\alpha)\,\mathrm{d} \alpha \gg \nu^{2}N^{\frac{77}{60}}. \end{aligned}$$
(3.10)

Proof

To prove (3.10), we write the left side as
$$\begin{aligned} \int_{\delta N}^{N} \int_{(\delta N)^{\frac{1}{2}}}^{N^{\frac{1}{2}}}\cdots \int_{(\delta N)^{\frac{1}{5}}}^{N^{\frac{1}{5}}} \int_{-\infty}^{+\infty} e \Biggl(\alpha \Biggl(\eta+\sum _{j=1}^{5}\lambda_{j}y_{j}^{j} \Biggr) \Biggr)K_{\nu }(\alpha)\,\mathrm{d} \alpha \,\mathrm{d}y_{1} \,\mathrm{d}y_{2}\cdots\,\mathrm{d}y_{5}, \end{aligned}$$
which, by (2.7), is
$$\begin{aligned} \int_{\delta N}^{N} \int_{(\delta N)^{\frac{1}{2}}}^{N^{\frac{1}{2}}}\cdots \int_{(\delta N)^{\frac{1}{5}}}^{N^{\frac{1}{5}}} \max \Biggl(0,\nu-\Biggl\vert \eta+ \sum_{j=1}^{5}\lambda_{j}y_{j}^{j} \Biggr\vert \Biggr) \,\mathrm{d}y_{1}\,\mathrm{d}y_{2}\cdots \,\mathrm{d}y_{5}. \end{aligned}$$
(3.11)
We let \(z_{k}=y_{k}^{k}\), \(k=1,2,3,4,5\), then the integral (3.11) can be written as
$$\begin{aligned} \frac{1}{120} \int_{\delta N}^{N} \cdots \int_{\delta N}^{N} z_{2}^{-\frac{1}{2}}z_{3}^{-\frac{2}{3}}z_{4}^{-\frac{3}{4}}z_{5}^{-\frac{4}{5}} \max \Biggl(0,\nu-\Biggl\vert \eta+\sum_{j=1}^{5} \lambda_{j}z_{j}\Biggr\vert \Biggr) \,\mathrm{d}z_{1} \cdots\,\mathrm{d}z_{5}. \end{aligned}$$
(3.12)
Since \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), and \(\lambda_{5}\) are not all of the same sign, we may assume without loss of generality that \(\lambda_{1}<0\), \(\lambda _{2}>0\). Consider the region
$$\mathcal{B}=\bigl\{ (z_{2},z_{3},z_{4},z_{5}): \delta^{\frac{1}{2}} N\leq z_{2} \leq 2\delta^{\frac{1}{2}} N, \delta N\leq z_{j} \leq2\delta N\ (j=3,4,5)\bigr\} . $$
Then, for δ sufficiently small and large N, whenever \((z_{2},z_{3},z_{4},z_{5})\in\mathcal{B}\) one has
$$2\delta N< -(\lambda_{2}z_{2}+\lambda_{3}z_{3}+ \lambda_{4}z_{4}+\lambda_{5}z_{5})\lambda _{1}^{-1}< \frac{1}{2}N $$
and so every \(z_{1}\) with \(|\lambda_{1}z_{1}+\cdots+\lambda_{5}z_{5}+\eta|\leq \frac{1}{2}\nu\) satisfies \(\delta N< z_{1}< N\). Therefore the integral (3.12) is greater than
$$\begin{aligned} \frac{1}{480}\nu^{2} \int_{\mathcal{B}}z_{2}^{-\frac{1}{2}}z_{3}^{-\frac {2}{3}}z_{4}^{-\frac{3}{4}} z_{5}^{-\frac{4}{5}} \,\mathrm{d}z_{2}\,\mathrm{d}z_{3} \,\mathrm{d}z_{4}\,\mathrm{d}z_{5} \gg\nu^{2}N^{\frac{77}{60}}. \end{aligned}$$
This completes the proof of Lemma 3.5. □
Together with Lemmas 3.3, 3.4, 3.5, we have
$$\begin{aligned} J(\mathfrak{C})= \int_{\mathfrak{C}}\prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta )K_{\nu}(\alpha)\,\mathrm{d}\alpha \gg \nu^{2}N^{\frac{77}{60}}. \end{aligned}$$
(3.13)

4 The intermediate region

Lemma 4.1

We have
$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{j}(\alpha)\bigr|^{2^{j}}K_{\nu}( \alpha)\,\mathrm{d} \alpha \ll N^{\frac{2^{j}}{j}-1+\varepsilon}, \quad j=2,3,4,5, \end{aligned}$$
(4.1)
$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{1}(\alpha)\bigr|^{2}K_{\nu}( \alpha)\,\mathrm{d} \alpha \ll NL. \end{aligned}$$
(4.2)

Proof

By (2.7), we have
$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{4}K_{\nu}( \alpha)\,\mathrm{d} \alpha \\ &\quad= \sum_{(\delta N)^{\frac{1}{2}}\leq p_{1},p_{2},p_{3},p_{4} \leq N^{\frac{1}{2}}} \prod _{i=1}^{4}\log p_{i} \max\bigl(0,\nu-\bigl| \lambda_{2}\bigl(p_{1}^{2}+p_{2}^{2}-p_{3}^{2}-p_{4}^{2} \bigr)\bigr|\bigr) \\ &\quad\ll L^{4}\sum_{(\delta N)^{\frac{1}{2}}\leq p_{1},p_{2},p_{3},p_{4} \leq N^{\frac{1}{2}}} \max\bigl(0, \nu-\bigl|\lambda_{2}\bigl(p_{1}^{2}+p_{2}^{2}-p_{3}^{2}-p_{4}^{2} \bigr)\bigr|\bigr). \end{aligned}$$
Since N is large, \(|\lambda_{2}(p_{1}^{2}+p_{2}^{2}-p_{3}^{2}-p_{4}^{2})|<\nu\) if and only if \(p_{1}^{2}+p_{2}^{2}=p_{3}^{2}+p_{4}^{2}\). Thus, by Hua’s inequality,
$$\begin{aligned} \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{4}K_{\nu}( \alpha)\,\mathrm{d} \alpha \ll\nu N^{1+\varepsilon}. \end{aligned}$$
The proofs of the cases \(j=3,4,5\) and (4.2) are similar. □

Lemma 4.2

We have
$$\begin{aligned} \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{2}\bigl|F_{4}( \alpha)\bigr|^{4}K_{\nu}(\alpha )\,\mathrm{d} \alpha \ll\nu N^{1+\varepsilon}. \end{aligned}$$
(4.3)

Proof

By (2.7), we have
$$\begin{aligned} & \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{2}\bigl|F_{4}( \alpha)\bigr|^{4}K_{\nu }(\alpha)\,\mathrm{d} \alpha \\ &\quad\ll L^{6}\mathop{\sum_{(\delta N)^{\frac{1}{2}}\leq p_{1},p_{2} \leq N^{\frac{1}{2}}}} _{(\delta N)^{\frac{1}{4}}\leq p_{3},p_{4},p_{5},p_{6} \leq N^{\frac{1}{4}}} \max\bigl(0,\nu-\bigl|\lambda_{2}\bigl(p_{1}^{2}-p_{2}^{2} \bigr)-\lambda _{4}\bigl(p_{3}^{4}+p_{4}^{4}-p_{5}^{4}-p_{6}^{4} \bigr)\bigr|\bigr) \\ &\quad\ll\nu L^{6}R(N), \end{aligned}$$
where \(R(N)\) is the number of the solutions of the equation
$$\begin{aligned} &\lambda_{2}\bigl(p_{1}^{2}-p_{2}^{2} \bigr)=\lambda_{4}\bigl(p_{3}^{4}+p_{4}^{4}-p_{5}^{4}-p_{6}^{4} \bigr), \\ &(\delta N)^{\frac{1}{2}}\leq p_{1},p_{2} \leq N^{\frac{1}{2}}, \qquad (\delta N)^{\frac{1}{4}}\leq p_{3},p_{4},p_{5},p_{6} \leq N^{\frac{1}{4}}. \end{aligned}$$
Then we have
$$R(N)\ll N^{\frac{1}{2}}\mathop{\sum_{(\delta N)^{\frac{1}{4}}\leq p_{3},p_{4},p_{5},p_{6} \leq N^{\frac{1}{4}}}}_{p_{3}^{4}+p_{4}^{4}-p_{5}^{4}-p_{6}^{4}= 0}1+ \mathop{\sum _{(\delta N)^{\frac{1}{4}}\leq p_{3},p_{4},p_{5},p_{6} \leq N^{\frac {1}{4}}}}_{p_{3}^{4}+p_{4}^{4}-p_{5}^{4}-p_{6}^{4}\neq0}d\bigl(\bigl|p_{3}^{4}+p_{4}^{4}-p_{5}^{4}-p_{6}^{4}\bigr| \bigr), $$
where \(d(n)\) is the divisor function. Now (4.3) follows from [3], (2.1). □

Lemma 4.3

([4])

Suppose that \((a,q)=1\), \(|\alpha -a/q|\leq q^{-2}\), then
$$\begin{aligned} \sum_{1\leq p \leq X}(\log p) e(p\alpha) \ll(\log X)^{5}\bigl(X^{1/2}q^{1/2}+X^{4/5}+Xq^{-1/2} \bigr). \end{aligned}$$

Lemma 4.4

([5])

Suppose that \((a,q)=1\), \(|\alpha -a/q|\leq q^{-2}\), \(\phi(x) =\alpha x^{k}+\alpha_{1}x^{k-1}+\cdots+\alpha_{k-1}x+\alpha_{k}\) (\(k\geq2\)), then
$$\begin{aligned} \sum_{1\leq p \leq X}(\log p) e\bigl(\phi(p)\bigr) \ll X^{1+\varepsilon}\bigl(q^{-1}+X^{-1/2}+qX^{-k} \bigr)^{4^{1-k}}. \end{aligned}$$

Lemma 4.5

For \(\tau<|\alpha|\leq P\), we have
$$\begin{aligned} V(\alpha):=\min\bigl(F_{1}(\alpha),F_{2}( \alpha)^{2}\bigr) \ll N^{1-\frac{\theta }{2}+\varepsilon}. \end{aligned}$$

Proof

Let \(\tau<|\alpha|\leq P\), we choose \(a_{j}\), \(q_{j}\) (\(j=1,2\)) so that \(|\lambda_{j}\alpha-a_{j}/q_{j}|\leq Q^{-1}q_{j}^{-1}\) with \((a_{j},q_{j})=1\) and \(1\leq q_{j} \leq Q\). By the method of Davenport and Heilbronn (see Lemma 11 of [6]), we have \(\max(q_{1},q_{2})\geq P\). Then Lemma 4.5 follows from Lemmas 4.3 and 4.4. □

Lemma 4.6

We have
$$\begin{aligned} J(\mathfrak{D})= \int_{\mathfrak{D}}\prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta )K_{\nu}(\alpha)\,\mathrm{d}\alpha \ll\nu^{2} N^{\frac{77}{60}-(\frac{\theta}{32}-\sigma)+\varepsilon}. \end{aligned}$$
(4.4)

Proof

By Lemmas 4.1, 4.2, 4.5, and Hölder’s inequality, we have
$$\begin{aligned} & \int_{\mathfrak{D}}\Biggl\vert \prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta)K_{\nu }(\alpha)\Biggr\vert \,\mathrm{d}\alpha \\ &\quad\ll V(\alpha)^{\frac{1}{16}} \int_{-\infty}^{+\infty}\Biggl\vert \bigl(F_{1}( \alpha)^{\frac {15}{16}}F_{2}(\alpha) +F_{1}( \alpha)F_{2}(\alpha)^{\frac{7}{8}} \bigr) \prod _{j=3}^{5}F_{j}(\alpha)\Biggr\vert K_{\nu}(\alpha)\,\mathrm{d}\alpha \\ &\quad\ll V(\alpha)^{\frac{1}{16}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{1}(\alpha)\bigr|^{2}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{15}{32}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{4}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{8}} \\ &\qquad{} \times \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)^{2}F_{4}( \alpha)^{4}\bigr| K_{\nu}(\alpha)\,\mathrm{d}\alpha \biggr)^{\frac{1}{4}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{3}(\alpha)\bigr|^{8}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{8}} \\ &\qquad{} \times \biggl( \int_{-\infty}^{+\infty}\bigl|F_{5}(\alpha)\bigr|^{32} K_{\nu}(\alpha)\,\mathrm{d}\alpha \biggr)^{\frac{1}{32}} +V(\alpha)^{\frac{1}{16}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{1}(\alpha)\bigr|^{2}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{2}}\\ &\qquad{}\times \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{4}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{3}{32}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)^{2}F_{4}( \alpha)^{4}\bigr| K_{\nu}(\alpha)\,\mathrm{d}\alpha \biggr)^{\frac{1}{4}}\\ &\qquad{} \times \biggl( \int_{-\infty}^{+\infty}\bigl|F_{3}(\alpha)\bigr|^{8}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{8}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{5}(\alpha)\bigr|^{32} K_{\nu}(\alpha)\,\mathrm{d}\alpha \biggr)^{\frac{1}{32}} \\ &\quad\ll\nu N^{\frac{77}{60}-\frac{\theta}{32}+\varepsilon} \ll\nu^{2} N^{\frac{77}{60}-(\frac{\theta}{32}-\sigma)+\varepsilon}. \end{aligned}$$
 □

5 The trivial region

Lemma 5.1

Let \(G(\alpha)=\sum e(\alpha f(x_{1},\ldots,x_{m}))\), 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
$$\begin{aligned} \int_{|\alpha|>A}\bigl|G(\alpha)\bigr|^{2}K_{\nu}(\alpha) \,\mathrm{d}\alpha \leq\frac{16}{A} \int_{-\infty}^{+\infty}\bigl|G(\alpha)\bigr|^{2}K_{\nu}( \alpha )\,\mathrm{d}\alpha. \end{aligned}$$
This is Lemma 2 of [7].

Lemma 5.2

We have
$$\begin{aligned} J(\mathfrak{c})= \int_{\mathfrak{c}}\prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta )K_{\nu}(\alpha)\,\mathrm{d}\alpha \ll\nu^{2} N^{\frac{77}{60}-(\theta-\sigma)+\varepsilon}. \end{aligned}$$

Proof

By Lemmas 4.1, 4.2, 5.1, and Hölder’s inequality, we have
$$\begin{aligned} & \int_{\mathfrak{c}} \Biggl|\prod_{j=1}^{5}F_{j}( \alpha)e(\alpha\eta)K_{\nu }(\alpha) \Biggr|\,\mathrm{d}\alpha \\ &\quad\ll\frac{1}{P} \int_{-\infty}^{+\infty} \prod_{j=1}^{5}\bigl|F_{j}( \alpha)\bigr|K_{\nu}(\alpha)\,\mathrm{d}\alpha \\ &\quad\ll\frac{1}{P}\max\bigl(\bigl|F_{5}(\alpha)\bigr|\bigr) \biggl( \int_{-\infty}^{+\infty}\bigl|F_{1}(\alpha)\bigr|^{2}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{2}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)\bigr|^{4}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{8}} \\ &\qquad{}\times \biggl( \int_{-\infty}^{+\infty}\bigl|F_{2}(\alpha)^{2}F_{4}( \alpha)^{4}\bigr| K_{\nu}(\alpha)\,\mathrm{d}\alpha \biggr)^{\frac{1}{4}} \biggl( \int_{-\infty}^{+\infty}\bigl|F_{3}(\alpha)\bigr|^{8}K_{\nu}( \alpha)\,\mathrm {d}\alpha \biggr)^{\frac{1}{8}} \\ &\quad\ll\nu N^{\frac{77}{60}-\theta+\varepsilon} \ll\nu^{2} N^{\frac{77}{60}-(\theta-\sigma)+\varepsilon}. \end{aligned}$$
 □

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), the Research Foundation of North China University of Water Conservancy and Electric Power (No. 201084).

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)
Department of Mathematics and Information Sciences, North China University of Water Resources and Electric Power
(2)
Department of Mathematics and Information Science, Henan University of Economics and Law

References

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Copyright

© Ge and Li 2016