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A PAIR OF EQUATIONS IN EIGHT PRIME CUBES AND POWERS OF 2

Published online by Cambridge University Press:  14 December 2022

XUE HAN
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, PR China e-mail: [email protected]
HUAFENG LIU*
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, PR China
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Abstract

In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$. In particular, we prove that $k=335$ is admissible, which improves the previous result.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

In 1951 and 1953, Linnik [Reference Linnik5, Reference Linnik6] showed that every large even integer N can be represented in the form of two primes and a bounded number of powers of 2, namely

(1.1) $$ \begin{align} N'=p_{1}+p_{2}+2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k'}}. \end{align} $$

Later, Liu et al. [Reference Liu, Liu and Wang8] proved that $k'=54000$ is acceptable in (1.1). After many improvements, up to now, the best result is $k'=8$ established by Pintz and Ruzsa [Reference Pintz and Ruzsa14]. In 2013, Kong [Reference Kong3] first considered the simultaneous representation of pairs of positive even integers as sums of two primes and powers of $2$ , that is,

$$ \begin{align*} \begin{cases} N_{1}'=p_{1}+p_{2}+2^{v_{1}}+2^{v_{2}}+\cdots +2^{v_{k'}}, \\ N_{2}'=p_{3}+p_{4}+2^{v_{1}}+2^{v_{2}}+\cdots +2^{v_{k'}}. \end{cases} \end{align*} $$

She proved that these equations are solvable for a pair of sufficiently large positive even integers $N_{1}'$ and $N_{2}'$ satisfying $N_{2}'\gg N_{1}'> N_{2}'$ for $k'=63$ unconditionally, and for $k'=31$ under the generalised Riemann hypothesis (GRH). Subsequently, Kong and Liu [Reference Kong and Liu4] improved the value of $k'$ to $34$ unconditionally and to $18$ under the GRH.

In 2001, based on the works of Linnik [Reference Linnik5, Reference Linnik6] and Gallagher [Reference Gallagher2], Liu and Liu [Reference Liu and Liu7] proved that every large even integer N can be written as a sum of eight cubes of primes and a bounded number of powers of $2$ , namely

$$ \begin{align*} N=p_{1}^{3}+p_{2}^{3}+\cdots+p_{8}^{3}+2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}}. \end{align*} $$

So far, the best result for this equation is $k=30$ obtained by Zhu [Reference Zhu19].

As a generalisation, in 2013, Liu [Reference Liu11] first considered the simultaneous representation of pairs of positive even integers $N_{1}$ and $N_{2}$ satisfying $N_{2}\gg N_{1}> N_{2}$ in the form

(1.2) $$ \begin{align} \begin{cases} N_{1}=p_{1}^{3}+p_{2}^{3}+\cdots+p_{8}^{3}+2^{v_{1}}+2^{v_{2}}+\cdots +2^{v_{k}}, \\ N_{2}=p_{9}^{3}+p_{10}^{3}+\cdots+p_{16}^{3}+2^{v_{1}}+2^{v_{2}}+\cdots +2^{v_{k}}, \end{cases} \end{align} $$

where k is a positive integer. Liu [Reference Liu11] proved that the equations in (1.2) are solvable for $k = 1432$ . This number k was improved successively to $ k=1364, k=658$ and $k=609$ by Platt and Trudgian [Reference Platt and Trudgian15], Zhao [Reference Zhao17] and Liu [Reference Liu9], respectively. We make a further improvement on the value of k in (1.2) by establishing the following result.

Theorem 1.1. For $k=335$ , the equations in (1.2) are solvable for every pair of sufficiently large positive even integers $N_{1}$ and $N_{2}$ satisfying $N_{2}\gg N_{1}>N_{2}$ .

To prove Theorem 1.1, we apply the circle method in combination with some new arguments of Kong and Liu [Reference Kong and Liu4]. To apply the circle method, similarly to [Reference Kong and Liu4], we divide $[0,1]^{2}$ into three arcs, which means we can avoid the limitation of two arcs in Liu [Reference Liu9] after applying integral transforms (see Section 4 for details), resulting in the sharper k in (1.2).

Notation 1.2. Throughout this paper, the letter p, with or without a subscript, always represents a prime. Both $N_{1}$ and $N_{2}$ denote sufficiently large positive even integers, $e(x)=\exp (2\pi ix)$ and $n\sim N$ means $N<n\leq 2N$ . The letter $\epsilon $ denotes a positive constant which is arbitrarily small but may not be the same at different occurrences.

2 Outline of the proof

In this section, we give an outline for the proof of Theorem 1.1. To apply the circle method, we let, for $i=1,2$ ,

$$ \begin{align*} P_{i}=N_{i}^{{1}/{9}-2\epsilon}, \quad Q_{i}=N_{i}^{{8}/{9}+\epsilon},\quad L=\frac{\log(N_{1}/\log N_{1})}{\log 2}. \end{align*} $$

For $i=1,2$ , we define the major arcs $\mathfrak {M}_{i}$ and minor arcs $C(\mathfrak {M}_{i})$ as

(2.1) $$ \begin{align} \mathfrak{M}_{i}=\bigcup_{1 \leq q_{i} \leq P_{i}}\bigcup_{\substack{1 \leq a_{i} \leq q_{i} \\ (a_{i}, q_{i})=1}}{ } \mathfrak{M}_{i}(a_{i}, q_{i}),\quad C(\mathfrak{M}_{i})=[0,1] \backslash \mathfrak{M}_{i}, \end{align} $$

where

$$ \begin{align*} \mathfrak{M}_{i}(a_{i}, q_{i})=\bigg\{\alpha_{i} \in[0,1]:\bigg|\alpha_{i}-\frac{a_{i}}{q_{i}}\bigg| \leq \frac{1}{q_{i} Q_{i}}\bigg\} \end{align*} $$

and

$$ \begin{align*} 1\leq a_{i}\leq q_{i}\leq Q_{i}, \quad (a_{i},q_{i})=1. \end{align*} $$

Note that the major arcs $\mathfrak {M}_{i}(a_{i}, q_{i})$ are mutually disjoint since $2P_{i}\leq Q_{i}$ . We further define

(2.2) $$ \begin{align} \mathfrak{M}=\mathfrak{M}_{1} \times \mathfrak{M}_{2}=\{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2}: \alpha_{1} \in \mathfrak{M}_{1}, \alpha_{2} \in \mathfrak{M}_{2}\}, \end{align} $$
(2.3) $$ \begin{align} C(\mathfrak{M})=[0,1]^{2} \backslash \mathfrak{M}. \end{align} $$

As in [Reference Ren16], let $\delta =10^{-4}$ and

$$ \begin{align*} U_{i}=\bigg(\frac{N_{i}}{16(1+\delta)}\bigg)^{{1}/{3}}, \quad V_{i}=U_{i}^{{5}/{6}}. \end{align*} $$

For $i=1,2$ , we set

(2.4) $$ \begin{align} S(\alpha_{i},U_{i})=\sum_{p \sim U_{i}}(\log p) e(p^{3} \alpha_{i}), \quad T(\alpha_{i},V_{i})=\sum_{p \sim V_{i}}(\log p) e(p^{3} \alpha_{i}), \end{align} $$
$$ \begin{align*} G(\alpha_{i})=\sum_{4 \leq v \leq L} e(2^{v} \alpha_{i}), \quad \mathscr{E}_{\lambda}=\{(\alpha_{1},\alpha_{2})\in[0,1]^{2}:|G(\alpha_{1}+\alpha_{2})|\geq \lambda L\}. \end{align*} $$

Let

$$ \begin{align*} R(N_{1},N_{2})=\sum\log p_{1}\log p_{2}\cdots\log p_{16} \end{align*} $$

be the weighted number of solutions of (1.2) in $(p_{1},p_{2},\ldots ,p_{16},v_{1},v_{2},\ldots ,v_{k})$ with

$$ \begin{align*} \begin{aligned} p_{1},p_{2},p_{3},p_{4} \sim U_{1}, \quad &p_{5},p_{6},p_{7},p_{8} \sim V_{1}, \\ p_{9},p_{10},p_{11},p_{12} \sim U_{2}, \quad &p_{13},p_{14},p_{15},p_{16} \sim V_{2},\\ 4 \leq v_{j} \leq L, \quad &j=1,2, \ldots, k. \end{aligned} \end{align*} $$

Then we rewrite $R(N_{1},N_{2})$ as

$$ \begin{align*} \begin{aligned} R(N_{1}, N_{2}) =&\,\bigg(\iint_{\mathfrak{M}}+\iint_{C(\mathfrak{M}) \cap \mathscr{E}_{\lambda}}+\iint_{C(\mathfrak{M}) \backslash \mathscr{E}_{\lambda}}\bigg) \,S^{4}(\alpha_{1}, U_{1})T^{4}(\alpha_{1}, V_{1})S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2}) \\ &\quad \times G^{k}(\alpha_{1}+\alpha_{2}) e(-\alpha_{1} N_{1}-\alpha_{2} N_{2})\,{d}\alpha_{1}\,{d}\alpha_{2} \\ :=&\,R_{1}(N_{1}, N_{2})+R_{2}(N_{1}, N_{2})+R_{3}(N_{1}, N_{2}). \end{aligned} \end{align*} $$

In Section 3, we first give some lemmas. In Section 4, we shall estimate $R_{i}(N_{1}, N_{2})$ for $i=1,2,3$ and complete the proof of Theorem 1.1.

3 Auxiliary lemmas

Let

(3.1) $$ \begin{align} C(q, a)=\sum_{\substack{m=1 \\ (m,q)=1}}^{q} e\bigg(\frac{a m^{3}}{q}\bigg), &\quad B(n,q)=\sum_{\substack{a=1\\ (a, q)=1}}^{q} C^{8}(q, a) e\bigg(-\frac{a n}{q}\bigg), \nonumber\\[-6pt] \\[-6pt]A(n, q)=\frac{B(n, q)}{ \varphi^{8}(q)},&\quad \mathfrak{S}(n)=\sum_{q=1}^{\infty} A(n, q). \nonumber\end{align} $$

Lemma 3.1. Let $\mathscr {A}(N_{i},k)=\{n_{i}\geq 2: n_{i}=N_{i}-2^{v_{1}}-2^{v_{2}}-\cdots -2^{v_{k}}\}$ with $k\geq 35$ . Then, for $N_{1} \equiv N_{2} \equiv 0\ ( \bmod \ 2)$ ,

$$ \begin{align*} \sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k)\\n_{2} \in \mathscr{A}(N_{2}, k)\\ n_{1} \equiv n_{2} \equiv 0 \pmod 2}} \mathfrak{S}(n_{1}) \mathfrak{S}(n_{2}) \geq 0.1596600336 L^{k}. \end{align*} $$

Proof. From (5.9) of [Reference Liu and Lü12] and Lemma 2.3 of [Reference Zhao and Ge18], for $p\geq 13 \text { and } p \equiv 1 \ ( \bmod \ 3)$ ,

$$ \begin{align*} 1+A(n,p)\geq 1-\frac{(2\sqrt{p}+1)^{8}}{(p-1)^{7}}, \end{align*} $$

and

$$ \begin{align*} \prod_{p \geq 17}(1+A(n_{i}, p))\geq 0.8206744593. \end{align*} $$

Then,

$$ \begin{align*} \begin{aligned} \prod_{p \geq 13}(1+A(n_{i}, p)) &=(1+A(n_{i}, 13)) \times \prod_{p \geq 17}(1+A(n_{i}, p)) \\ &\geq 0.4233091149\times 0.8206744593\\ &\geq 0.3473989790:=C. \end{aligned} \end{align*} $$

Noting that $\mathfrak {S}(n_{i})=2(1-{1}/{2^{8}})\prod _{p>3}(1+A(n_{i},p))$ and putting $q=\prod _{3<p<12}=385$ ,

$$ \begin{align*} \begin{aligned} \sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2)}} & \mathfrak{S}(n_{1}) \mathfrak{S}(n_{2})\\ &\geq\bigg(2\bigg(1-\frac{1}{2^{8}}\bigg) C\bigg)^{2} \sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2)}} \prod_{i=1}^{2} \prod_{3<p_{i}<12}(1+A(n_{i}, p_{i}))\\ &\geq\bigg(2\bigg(1-\frac{1}{2^{8}}\bigg) C\bigg)^{2} \sum_{\substack{1 \leq j_{1} \leq q}} \sum_{\substack{1 \leq j_{2} \leq q}}\sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2) \\ n_{1} \equiv j_{1}\ (\!\!\bmod q)\\n_{2} \equiv j_{2}\ (\!\!\bmod q)}}\prod_{i=1}^{2} \prod_{3<p_{i}<12}(1+A(\,j_{i}, p_{i}))\\ &\geq\bigg(2\bigg(1-\frac{1}{2^{8}}\bigg) C\bigg)^{2} \sum_{1 \leq j_{1} \leq q} \sum_{1 \leq j_{2} \leq q}\prod_{i=1}^{2} \prod_{3<p_{i}<12}(1+A(\,j_{i}, p_{i})) \sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k)\\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2)\\n_{1} \equiv j_{1}\ (\!\!\bmod q)\\n_{2} \equiv j_{2}\ (\!\!\bmod q)}} 1. \end{aligned} \end{align*} $$

Considering the inner sum,

$$ \begin{align*} S:=\sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k)\\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2)\\n_{1} \equiv j_{1}\ (\!\!\bmod q)\\n_{2} \equiv j_{2}\ (\!\!\bmod q)}} 1=\sum_{\substack{4 \leq v_{j} \leq L, 1 \leq j \leq k, i=1,2\\2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{i}\ (\!\!\bmod 2)\\2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{i}-j_{i}\ (\!\!\bmod q)}}1. \end{align*} $$

Since $N_{1} \equiv N_{2} \equiv 0\ (\bmod \ 2)$ ,

$$ \begin{align*} \begin{cases} 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1} \ (\bmod \ 2) \\ 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{2} \ (\bmod \ 2) \end{cases} \end{align*} $$

is equivalent to

$$ \begin{align*} 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1} \ (\bmod \ 2). \end{align*} $$

Additionally, if $N_{2} \equiv N_{1}+t\ (\bmod \ q)$ and $j_{2} \equiv j_{1}+t\ (\bmod \ q)$ with $1\leq t\leq q$ ,

$$ \begin{align*} \begin{cases} 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1}-j_{1} \ (\bmod \ q)\\ 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{2}-j_{2} \ (\bmod \ q)\end{cases} \end{align*} $$

is equivalent to

$$ \begin{align*} 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1}-j_{1} \ (\bmod \ q). \end{align*} $$

Therefore, when $N_{1} \equiv N_{2} \equiv 0\ (\bmod \ 2)$ , $N_{2} \equiv N_{1}+t\ (\bmod \ q)$ and $j_{2} \equiv j_{1}+t\ (\bmod \ q)$ ,

$$ \begin{align*} \begin{aligned} S \geq \sum_{\substack{4 \leq v_{1}, v_{2}, \ldots, v_{k} \leq L \\ 2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1}\ (\!\!\bmod 2)\\2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv N_{1}-j_{1}\ (\!\!\bmod q)}} \bigg(\frac{L}{\rho(3 q)}+O(1)\bigg)^{k} \sum_{\substack{4 \leq v_{1},v_{2},\ldots, v_{k} \leq \rho(3 q)\\2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}} \equiv a_{j}\ (\!\!\bmod 3 q)}} 1, \end{aligned} \end{align*} $$

where the natural number $a_{j}\in [1,3q]$ satisfies the conditions $a_{j} \equiv N_{1} \ (\bmod \ 3)$ and $a_{j} \equiv N_{1}-j_{1} \ (\bmod \ q)$ , and $\rho (q)$ denotes the smallest positive integer $\rho $ such that $2^{\rho }\equiv 1 \ (\bmod \ q)$ .

Noting that

$$ \begin{align*} S \geq \frac{1}{3 q}\bigg(\frac{L}{\rho(3 q)}+O(1)\bigg)^{k} \sum_{t=0}^{3 q-1} e\bigg(\frac{t a_{j}}{3q}\bigg)\bigg(\sum_{1 \leq s \leq\rho(3 q)} e\bigg(\frac{t 2^{s}}{3 q}\bigg)\bigg)^{k}, \end{align*} $$

we get

$$ \begin{align*} \begin{aligned} S & \geq \frac{1}{3 q}\bigg(\frac{L}{\rho(3 q)}+O(1)\bigg)^{k}(\,\rho(3 q)^{k}-(3 q-1)(\max )^{k}) \\ &=\frac{L^{k}}{3 q}\bigg(1-(3 q-1)\bigg(\frac{\max }{\rho(3 q)}\bigg)^{k}\bigg)+O(L^{k-1}), \end{aligned} \end{align*} $$

where

$$ \begin{align*} \max =\max \bigg\{\bigg|\sum_{1 \leq s \leq \rho(3 q)} e\bigg(\frac{j 2^{s}}{3 q}\bigg)\bigg|: 1 \leq j \leq 3 q-1\bigg\}. \end{align*} $$

Since $3q=1155$ and $\rho (3q)=60$ , with the help of a computer,

$$ \begin{align*} \max =30\ldots, \quad(3 q-1)\bigg(\frac{\max }{\rho(3 q)}\bigg)^{50}<10^{-10}. \end{align*} $$

Therefore,

$$ \begin{align*} S \geq \frac{(1-10^{-10}) L^{k}}{3 q}+O(L^{k-1}). \end{align*} $$

By a numerical calculation,

$$ \begin{align*} \max _{1 \leq t \leq q}\bigg(\sum_{1 \leq j_{1} \leq q} \prod_{3<p_{1}<12}(1+A(\,j_{1}, p_{1})) \prod_{3<p_{2}<12}(1+A(\,j_{1}+t, p_{2}))\bigg) \geq 384.9999769. \end{align*} $$

Then,

$$ \begin{align*} \sum_{\substack{n_{1} \in \mathscr{A}(N_{1}, k) \\ n_{2} \in \mathscr{A}(N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0\ (\!\!\bmod 2)}} \mathfrak{S}(n_{1}) \mathfrak{S}(n_{2}) &\geq 384.9999769\bigg(2\bigg(1-\frac{1}{2^{8}}\bigg)C\bigg)^{2}\frac{(1-10^{-10}) L^{k}}{3 q}\\ & \geq 0.1596600336L^{k}.\\[-2.8pc] \end{align*} $$

Lemma 3.2 [Reference Liu and Lü12, Lemma 2.1].

Let $\mathfrak {M}_{i}$ , $S(\alpha _{i},U_{i})$ and $T(\alpha _{i},V_{i})$ be defined as in (2.1) and (2.4), respectively. For ${N_{i}}/{2}\leq n_{i}\leq N_{i}$ ,

$$ \begin{align*} \int_{\mathfrak{M}_{i}} S^{4}(\alpha_{i},U_{i}) T^{4}(\alpha_{i},V_{i}) e(-n_{i} \alpha_{i}) \,{d} \alpha_{i}=\frac{1}{3^{8}} \mathfrak{S}(n_{i})\mathfrak{J}(n_{i})+O(N_{i}^{{13}/{9}} L^{-1}), \end{align*} $$

where $\mathfrak {S}(n_{i})$ is defined as in (3.1) and satisfies $\mathfrak {S}(n_{i})\gg 1$ for $n_{i} \equiv 0\ (\bmod \ 2)$ , and $\mathfrak {J}(n_{i})$ is defined as

$$ \begin{align*} \mathfrak{J}(n_{i}):=\sum_{\substack{m_{1}+m_{2}+\cdots+m_{8}=n_{i}\\U_{i}^{3}<m_{1},m_{2},m_{3},m_{4}\leq 8U_{i}^{3}\\V_{i}^{3}<m_{5},m_{6},m_{7},m_{8}\leq 8V_{i}^{3}}}(m_{1}m_{2} \ldots m_{8})^{-{2}/{3}} \end{align*} $$

and satisfies $N_{i}^{{13}/{9}}\ll \mathfrak {J}(n_{i})\ll N_{i}^{{13}/{9}}$ .

Lemma 3.3 [Reference Zhao and Ge18, Lemma 2.6].

For $(1-\delta )N_{i}\leq n_{i}\leq N_{i}$ ,

$$ \begin{align*} \mathfrak{J}(n_{i})>1.42432055N_{i}^{{13}/{9}}. \end{align*} $$

Lemma 3.4. We have $\operatorname {meas}(\mathscr {E}_{\lambda }) \ll N_{i}^{-E(\lambda )}$ with $E(0.9570253)>\tfrac 89+10^{-10}$ .

Proof. This is (2.7) in Lemma 2.1 of Zhao [Reference Zhao17].

Lemma 3.5 [Reference Zhao17, Lemma 2.5].

Let $\mathfrak {M}_{i}$ and $S(\alpha _{i},U_{i})$ be defined as in (2.1) and (2.4), respectively. We have

$$ \begin{align*} \max _{\alpha_{i} \in C(\mathfrak{M}_{i})}|S(\alpha_{i},U_{i})| \ll N_{i}^{{11}/{36}+\epsilon}. \end{align*} $$

Lemma 3.6. Let $S(\alpha _{i},U_{i})$ and $T(\alpha _{i},V_{i})$ be defined as in (2.4). We have

$$ \begin{align*} \int_{0}^{1}|S(\alpha_{i},U_{i}) T(\alpha_{i},V_{i})|^{4} \,{d} \alpha_{i} \leq 0.134694091 N_{i}^{{13}/{9}}. \end{align*} $$

Proof. The idea of the proof is similar to that of Lemma 2.6 in Liu and Lü [Reference Liu and Lü13]. However, we take $\nu =100552$ obtained by Elsholtz and Schlage-Puchta [Reference Elsholtz and Schlage-Puchta1] instead of $147185.22$ obtained by Liu [Reference Liu10]. This leads to a better upper bound.

Here we only consider the case $i=1$ since the case $i=2$ can be proved similarly. From (2.7) of Ren [Reference Ren16] and Proposition 2 of Elsholtz and Schlage-Puchta [Reference Elsholtz and Schlage-Puchta1],

$$ \begin{align*} \sum_{{N_{1}}/{9}<l\leq N_{1}}r^{2}(l)\leq \vartheta(0)\leq (\nu+o(1))U_{1}V_{1}^{4}L^{-8}, \end{align*} $$

where $\nu {\kern-1.2pt}={\kern-1.2pt}100552$ , $r(n)$ denotes the number of representations of n as $p_{1}^{3}+p_{2}^{3}+ p_{3}^{3}+p_{4}^{3}$ with $p_{1},p_{2}\sim U_{1}$ , $p_{3},p_{4}\sim V_{1}$ and $\vartheta (0)$ denotes the number of solutions of the equation $p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}=p_{5}^{3}+p_{6}^{3}+p_{7}^{3}+p_{8}^{3}$ with $p_{1},p_{2},p_{5},p_{6} \sim U_{1}, p_{3},p_{4},p_{7}, p_{8} \sim V_{1}$ .

Therefore,

$$ \begin{align*} \begin{split} \int_{0}^{1}|S(\alpha_{1},U_{1}) T(\alpha_{1},V_{1})|^{4} \,{d} \alpha_{1} &\leq (\log (2U_{1}))^{4}(\log(2V_{1}))^{4}\vartheta(0)\\ &\leq 0.134694091N_{1}^{{13}/{9}}. \\[-1.7pc] \end{split} \end{align*} $$

4 Proof of Theorem 1.1

To prove Theorem 1.1, we first estimate $R_{1}(N_{1},N_{2})$ . By Lemmas 3.1, 3.2 and 3.3,

(4.1) $$ \begin{align} R_{1}(N_{1}, N_{2}) & =\iint_{\mathfrak{M}} S^{4}(\alpha_{1}, U_{1})T^{4}(\alpha_{1}, V_{1})S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2}) \nonumber\\[4pt] &\quad \times G^{k}(\alpha_{1}+\alpha_{2}) e(-\alpha_{1} N_{1}-\alpha_{2} N_{2}) \,{d} \alpha_{1} \,{d} \alpha_{2} \nonumber\\[4pt] &\geq \bigg(\frac{1}{3^{8}}\bigg)^{2}\sum_{\substack{n_{1}\in\mathscr{A}(N_{1},k)\\n_{2}\in\mathscr{A}(N_{2},k)}}\mathfrak{S}(n_{1})\mathfrak{S}(n_{2}) \mathfrak{J}(n_{1})\mathfrak{J}(n_{2})\\[4pt] &\geq \frac{0.1596600336\times(1.42432055)^{2}}{3^{16}}(N_{1}N_{2})^{{13}/{9}}L^{k}\nonumber\\[4pt] &\geq 7.524395606\times10^{-9}(N_{1}N_{2})^{{13}/{9}}L^{k},\nonumber \end{align} $$

where $\mathfrak {M}$ is defined by (2.2).

Next, we estimate $R_{2}(N_{1},N_{2})$ . By (2.1) and (2.3),

$$ \begin{align*} \begin{aligned} C(\mathfrak{M}) &\subset \{(\alpha_{1}, \alpha_{2}): \alpha_{1} \in C(\mathfrak{M}_{1}), \alpha_{2} \in[0,1]\} \cup\{(\alpha_{1}, \alpha_{2}): \alpha_{1} \in[0,1], \alpha_{2} \in C(\mathfrak{M}_{2})\}. \end{aligned} \end{align*} $$

From Lemma 3.5 and the trivial bounds of $G(\alpha _i)$ and $T(\alpha _{i}, V_{i})$ ,

(4.2) $$ \begin{align} \kern-10pt\begin{aligned} R_{2}(N_{1}, N_{2}) &=\iint_{C(\mathfrak{M}) \cap \mathscr{E}_{\lambda}}S^{4}(\alpha_{1}, U_{1})T^{4}(\alpha_{1}, V_{1})S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2}) \\&\qquad \times G^{k}(\alpha_{1}+\alpha_{2}) e(-\alpha_{1} N_{1}-\alpha_{2} N_{2}) \,{d} \alpha_{1} \,{d} \alpha_{2}\\&\ll L^{k}\bigg(\iint_{\substack{(\alpha_{1}, \alpha_{2}) \in C(\mathfrak{M}_{1}) \times[0,1] \\ |G(\alpha_{1}+\alpha_{2})| \geq \lambda L}}+\iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1] \times C(\mathfrak{M}_{2}) \\ |G(\alpha_{1}+\alpha_{2})| \geq \lambda L}}\bigg)\\&\qquad S^{4}(\alpha_{1}, U_{1})T^{4}(\alpha_{1}, V_{1})S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2})\,{d} \alpha_{1} \,{d} \alpha_{2} \\&\ll L^{k} N_{1}^{{10}/{9}}N_{1}^{{11}/{9}+\epsilon} \iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2} \\ |G(\alpha_{1}+\alpha_{2})| \geq\lambda L}}|S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2})| \,{d} \alpha_{1} \,{d} \alpha_{2} \\&\qquad +L^{k} N_{2}^{{10}/{9}}N_{2}^{{11}/{9}+\epsilon} \iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2} \\ |G(\alpha_{1}+\alpha_{2})| \geq \lambda L}}|S^{4}(\alpha_{1},U_{1})T^{4}(\alpha_{1}, V_{1})| \,{d} \alpha_{1} \,{d} \alpha_{2}. \end{aligned} \end{align} $$

Let $\varpi =\alpha _{1}+\alpha _{2}$ . By the periodicity of $G(\alpha )$ ,

$$ \begin{align*} \begin{split} \iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2} \\ |G(\alpha_{1}+\alpha_{2})| \geq \lambda L}} &|S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2})| \,{d} \alpha_{1} \,{d} \alpha_{2}\\ &=\int_{0}^{1}|S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2})|\bigg(\int_{\substack{\varpi \in[\alpha_{2}, 1+\alpha_{2}] \\|G(\varpi)| \geq \lambda L}} \,{d} \varpi\bigg) \,{d} \alpha_{2}. \end{split} \end{align*} $$

By Lemmas 3.4 and 3.6,

(4.3) $$ \begin{align} \iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2} \\|G(\alpha_{1}+\alpha_{2})| \geq \lambda L}}|S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2})| \,{d} \alpha_{1} \,{d} \alpha_{2} \ll N_{2}^{{13}/{9}}N_{1}^{-{8}/{9}-10^{-10}}. \end{align} $$

Similarly,

(4.4) $$ \begin{align} \iint_{\substack{(\alpha_{1}, \alpha_{2}) \in[0,1]^{2} \\|G(\alpha_{1}+\alpha_{2})| \geq \lambda L}}|S^{4}(\alpha_{1},U_{1})T^{4}(\alpha_{1}, V_{1})| \,{d} \alpha_{1} \,{d} \alpha_{2} \ll N_{1}^{{13}/{9}}N_{2}^{-{8}/{9}-10^{-10}}. \end{align} $$

From (4.2)–(4.4),

(4.5) $$ \begin{align} R_{2}(N_{1},N_{2})&\ll N_{1}^{{10}/{9}}N_{1}^{{11}/{9}+\epsilon}N_{2}^{{13}/{9}}N_{1}^{-{8}/{9}-10^{-10}}L^{k}+ N_{2}^{{10}/{9}}N_{2}^{{11}/{9}+\epsilon}N_{1}^{{13}/{9}}N_{2}^{-{8}/{9}-10^{-10}}L^{k} \nonumber\\ &\ll (N_{1}N_{2})^{{13}/{9}}L^{k-1}, \end{align} $$

where $N_{2}\gg N_{1}>N_{2}$ .

Finally, we estimate $R_{3}(N_{1},N_{2})$ . By Lemma 3.6 and the definition of $\mathscr {E}_{\lambda }$ ,

(4.6) $$ \begin{align} \begin{aligned} R_{3}& (N_{1}, N_{2}) \\ & = \iint_{C(\mathfrak{M}) \backslash \mathscr{E}_{\lambda}}S^{4}(\alpha_{1}, U_{1})T^{4}(\alpha_{1}, V_{1})S^{4}(\alpha_{2},U_{2})T^{4}(\alpha_{2}, V_{2}) \\ &\quad \times G^{k}(\alpha_{1}+\alpha_{2}) e(-\alpha_{1} N_{1}-\alpha_{2} N_{2}) \,{d} \alpha_{1} \,{d} \alpha_{2} \\ & \leq(\lambda L)^{k} \int_{0}^{1} |S^{4}(\alpha_{1},U_{1}) T^{4}(\alpha_{1},V_{1})| \,{d} \alpha_{1}\int_{0}^{1} |S^{4}(\alpha_{2},U_{2}) T^{4}(\alpha_{2},V_{2})| \,{d} \alpha_{2} \\ & \leq 0.0181424982 \lambda^{k}(N_{1} N_{2})^{{13}/{9}} L^{k}. \end{aligned} \end{align} $$

Putting (4.1), (4.5) and (4.6) together,

$$ \begin{align*} \begin{aligned} R(N_{1}, N_{2})&>R_{1}(N_{1}, N_{2})-R_{3}(N_{1}, N_{2})+O((N_{1} N_{2})^{{13}/{9}} L^{k-1}) \\ &>(7.524395606\times10^{-9}-0.0181424982\lambda^{k})(N_{1} N_{2})^{{13}/{9}} L^{k}, \end{aligned} \end{align*} $$

where $\lambda =0.9570253$ . Then we can deduce that

$$ \begin{align*}R(N_{1}, N_{2})>0\end{align*} $$

provided that $k\geq 335$ . Thus, we complete the proof of Theorem 1.1.

Acknowledgement

The authors would like to thank the referee for useful comments.

Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 12171286).

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