A Diophantine approximation problem with unlike powers of primes

1.   Introduction

In 1953, Prachar ^[1] demonstrated that any sufficiently large odd integer $N$ can be expressed as

where ${p}_{1}$, ${p}_{2}$, ${p}_{3}$, ${p}_{4}$, and ${p}_{5}$ are prime numbers. Subsequently, as a result of [2, Theorem 1], Ren and Tsang reached the same conclusion as Prachar. It is significant to explore the analogous formulation for Diophantine inequalities. In 1973, Vaughan ^[3] first proved that for any real $\eta$, there are infinitely many solutions in primes ${{p}_{j}}$ to the inequality

with $\xi = {1}/{10}$. The exponent was subsequently improved several times, and the best result up to now is due to K. Matomäki ^[4] with $\xi = {2}/{9}$. Moreover, it is worth mentioning that Diophantine inequality (1.1) has also been resolved for special prime numbers. The solvability of (1.1) with primes ${{p}_{1}}$, ${{p}_{2}}$, and ${{p}_{3}}$, where ${{p}_{i}}+2$ are almost-primes, is discussed in ^[5]. Similarly, the cases where ${{p}_{3}} = {{x}^{2}}+{{y}^{2}}+1$ or ${{p}_{i}} = \left[{{n}^{c}} \right]$ (Piatetski-Shapiro primes) are addressed in ^[6,7], respectively.

Then for related results for Diophantine inequalities with unlike powers of primes, we consider the following examples. Let ${{\lambda }_{1}}$, ${{\lambda }_{2}}$, ${{\lambda }_{3}}$, ${{\lambda }_{4}}$, and ${{\lambda }_{5}}$ be non-zero real numbers, not all negative, and suppose that ${{{\lambda }_{1}}}/{{{\lambda }_{2}}}\;$ is irrational. In 2016, Ge and Li ^[8] demonstrated that for any given real numbers $\eta$ and $\sigma$ with $0 < \sigma < \frac{1}{720}$, there exist infinitely many solutions in prime numbers ${{p}_{j}}$ to the inequality

In 2017, Mu ^[9] made further improvements by extending the range of the exponent to $\sigma \le \frac{1}{180}$ using the method outlined in Languasco and Zaccagnini ^[10]. Subsequently, Liu ^[11] optimized the result further, obtaining $\sigma \le \frac{5}{288}$. Inspired by the work of Wang and Yao ^[12], Mu and Qu ^[13] combined the sieve methods from Harman ^[14] and Harman and Kumchev ^[15] to refine Liu's result and prove that (1.2) holds for $\sigma \le \frac{5}{252}$. Very recently, this result was improved by Zhu ^[16], who obtained $\sigma \le \frac{1}{48}$. In 2022, Zhu ^[17] employed an innovative approach to estimate the correlation integral over the minor arcs, leading to a more robust estimation that $\sigma \le \frac{19}{756}$.

Building upon the advancements made in the study of Diophantine inequalities, subsequent research has explored different variations and forms of these inequalities. For clarity in discussion, we first introduce the definition of a well-spaced sequence. A set of positive real numbers $\mathcal{V}$ is called a well-space sequence if there exists a constant $c > 0$ such that

On this basis, in 2018, Ge and Zhao ^[18] dropped the linear prime variable in (1.2) and considered the exceptional set for the inequality

where ${{\lambda }_{2}}$, ${{\lambda }_{3}}$, ${{\lambda }_{4}}$, and ${{\lambda }_{5}}$ are non-zero real numbers, not all negative. Let $\mathcal{D}$ be a well-spaced sequence, and suppose $\delta > 0$, and ${{{\lambda }_{1}}}/{{{\lambda }_{2}}}\;$ is algebraic and irrational. Denote by $E(\mathcal{D}, X, \delta)$ the number of $v\in \mathcal{D}$ with $v\le X$ such that the inequality (1.3) has no solution in primes ${{p}_{2}}$, ${{p}_{3}}$, ${{p}_{4}}$, and ${{p}_{5}}$. Recently, Ge and Zhao ^[18] demonstrated that

Subsequently, Mu and Gao ^[19], as well as Liu and Liu ^[20], further improved upon the result of Ge and Zhao.

In this paper, we drop the final prime variable in (1.2) and examine the exceptional set for the inequality

Drawing on techniques from ^[18,20], we establish the following results.

Theorem 1.1. Let $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ be non-zero real numbers, not all negative. Suppose that ${{{\lambda }_{1}}}/{{{\lambda }_{3}}}\;$is irrational and algebraic, $\delta > 0$, and the set $\mathcal{V}$ is a well-spaced sequence. Denote by $E(\mathcal{V}, X, \delta)$ the number of $v\in \mathcal{V}$ with $v\le X$ such that the inequality

has no solution in primes ${{p}_{1}}$, ${{p}_{2}}$, ${{p}_{3}}$, ${{p}_{4}}$. Then, for any $\varepsilon > 0$, we have

Theorem 1.2. Let $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ be non-zero real numbers, not all negative. Suppose that ${{{\lambda }_{1}}}/{{{\lambda }_{3}}}\;$is irrational and algebraic, $\delta > 0$, and the set $\mathcal{V}$ is a well-spaced sequence. Then there is a sequence ${{X}_{j}}\to \infty$ such that

for any $\varepsilon > 0$. Moreover, if the convergent denominators ${{q}_{j}}$ for ${{{\lambda }_{1}}}/{{{\lambda }_{3}}}\;$satisfy $q_{j+1}^{1-w}\ll {{q}_{j}}$ for $w\in \left[0 \right., \left. 1 \right)$, then, for all $X\ge 1$ and any $\varepsilon > 0$,

with

Remark. Theorem 1.1 follows directly from Theorem 1.2. Since ${{{\lambda }_{1}}}/{{{\lambda }_{3}}}\;$ is algebraic, we can set $w = \varepsilon$, and thus $\chi = \frac{1}{12}$. Therefore, we focus on proving Theorem 1.2 in the following.

Notation. Throughout this paper, the letter $p$ denotes a prime number, with or without subscripts. The non-zero real numbers ${{\lambda }_{1}}$, ${{\lambda }_{2}}$, ${{\lambda }_{3}}$, ${{\lambda }_{4}}$, and $\delta$ are given constants. Implicit constants in the $O$, $\ll$ and $\gg$ notations usually depend at most on ${{\lambda }_{1}}$, ${{\lambda }_{2}}$, ${{\lambda }_{3}}$, and ${{\lambda }_{4}}$. The letter $\varepsilon$ represents an arbitrarily small positive constant, but not necessarily the same each time. We write $e(x) = \exp (2\pi ix)$.

2.   Outline of the method

Let $X$ be a sufficiently large positive number. Suppose that ${{I}_{k}} = \left[{{(\eta X)}^{{1}/{k}\; }}, {{(X)}^{{1}/{k}\; }} \right]$ for $k = 1, 2, 3, 4$ and $0 < \tau < 1$. Define

and

for $k = 1, 2, 3, 4$. Then, it is simple to obtain

For convenience, $\Phi$ is defined as an arbitrary measurable subset of $\mathbb{R}$, and then

By (2.3), we have

Thus

where $\mathcal{N}(v, X)$ counts the number of solutions to the inequality

To estimate (2.4), we divide the region of integration into three parts: The major arc $\mathcal{M}$, the minor arc $m$, and the trivial arc $t$, defined as

and then we can write

where $\phi = {{X}^{\frac{5}{24}-1-\varepsilon }}, \xi = {{X}^{-\frac{7}{9}-\varepsilon }}, \gamma = {{\tau }^{-2}}{{X}^{\frac{7}{12}+2\varepsilon }}$.

It is sufficient to establish a positive lower bound for $\mathcal{I}(v, X; \mathbb{R})$. We employ a standard dyadic argument, focusing on those values of $v$ that satisfy $\frac{1}{2}X\le v\le X$. We will estimate the three terms on the right-hand side of (2.5) in Sections 4–6. Additionally, Sections 3 and 6 will present preliminary lemmas and conclude the proof of Theorem 1.2.

3.   Preliminary lemmas

Lemma 3.1. Suppose that $X\ge {{Z}_{1}}\ge {{X}^{\frac{4}{5}+2\varepsilon }}$ and $\left| {{S}_{1}}({{\lambda }_{1}}\alpha) \right| > {{Z}_{1}}$. Then, there exist two coprime integers $a$ and $q$ such that

$Proof$. This is extracted from [13, Lemma 4.1].

Lemma 3.2. Suppose that $\alpha$ is a real number, and there exist $a\in \mathbb{Z}$ and $q\in \mathbb{N}$ with

Then one has

$Proof$. See [18, Lemma 2.3].

Lemma 3.3. Let $k\ge 1$ be a real number. For any fixed real number $A\ge 6$, we have

$Proof$. This follows from [21, Corollary 2.3].

Lemma 3.4. Let $k$ be a positive integer, we have

$Proof$. This lemma corresponds to [13, Lemma 3.3] or [19, Lemma 3.1].

Lemma 3.5. Suppose ${{X}^{\frac{1}{8}-1}}\le \left| \alpha \right|\le {{X}^{-\frac{1}{8}}}$, we have

$Proof$. This result is derived from [22, Theorem 1] by taking $q = [{{\left| {{\lambda }_{4}}\alpha \right|}^{-1}}]$ and $a = 1$. It can also be found in ^[20].

Lemma 3.6. For $k = 1, 2, 3, 4$, and a positive integer $m$ such that $1\le m\le k$, we have

$Proof$. These results are derived from Hua's Lemma, with further details outlined in [3, Lemma 2.5].

Lemma 3.7. Suppose that $\alpha$ is a real number, and there exist $a\in \mathbb{Z}$ and $q\in \mathbb{N}$ with

Then one has

$Proof$. This result is derived from [23, Theorem 1] by taking $k = 4$.

Lemma 3.8. Let $m(3) = \frac{1}{36}$, $m(4) = \frac{1}{96}$, $\lambda$ and $\mu$ be nonzero constants. For $i = 3, 4$, suppose that ${{X}^{\frac{1}{i}-m(i)+\varepsilon }}\le {{Z}_{i}}\le {{X}^{\frac{1}{i}}}$. We define

i) For $\alpha \in \rho \left({{Z}_{i}} \right)$, there exist coprime integers ${{a}_{i}}$, ${{q}_{i}}$ satisfying

ii) For $l\ge4$, we have

$Proof$. For $i = 3$, let ${P} = {{X}^{\frac{1}{3}}}$, ${Q} = {{P}^{\frac{3}{2}}}$. By Dirichlet's approximation theorem, there exist two coprime integers ${{a}_{3}}$, ${{q}_{3}}$ with $1\le {{q}_{3}}\le {Q}$ and $\left| \alpha {{\lambda }_{3}}{{q}_{3}}-{{a}_{3}} \right|\le {{Q}^{-1}}$. Then it follows from Lemma 3.2 and the hypothesis ${{Z}_{3}}\ge {{X}^{\frac{1}{3}-m(3)+\varepsilon }}$, we have

Thus, we have

For $i = 4$, let ${P} = {{X}^{\frac{1}{4}}}$, ${Q} = {{P}^{\frac{47}{21}}}$. By Dirichlet's approximation theorem, there exist two coprime integers ${{a}_{4}}$, ${{q}_{4}}$ with $1\le {{q}_{4}}\le {Q}$ and $\left| \alpha {{\lambda }_{4}}{{q}_{4}}-{{a}_{4}} \right|\le {{Q}^{-1}}$. Then it follows from Lemma 3.7 and the hypothesis ${{Z}_{4}}\ge {{X}^{\frac{1}{4}-m(4)+\varepsilon }}$, we have

Thus, we have

which completes the proof of Lemma 3.8 i).

Next, we give the proof of Lemma 3.8 ii). Using Brüdern's method from ^[24], the result can be established for the case when ${\lambda} = 1$. To simplify the analysis, we assume that $\lambda = 1$. Define ${{Q}_{i}} = {{X}^{\frac{2}{i}+\varepsilon }}Z_{i}^{-2}$ and

Let ${{V}_{i}}\left(\alpha \right)$ be a function of period 1. Further, define the function ${{V}_{i}}(\alpha)$ for $\alpha \in \left({{Q}_{i}}{{X}^{-1}}, 1+{{Q}_{i}}{{X}^{-1}} \right]$ as

where ${{\rho }_{i}}^{\prime }$ is the union of all intervals ${{\rho }_{i}}^{\prime }\left({{q}_{i}}, {{a}_{i}} \right)$ such that $1\le {{a}_{i}}\le {{q}_{i}}\le {{Q}_{i}}$ and $\left({{a}_{i}}, {{q}_{i}} \right) = 1$. Let $\rho _{i}^{*} = {{\rho }_{i}}^{\prime }+\mathbb{Z}$ be the union of all such intervals, then

For $\alpha \in \rho \left({{Z}_{i}} \right)$, by combining (3.1)–(3.3), we obtain

Next, we express

where $l\ge 4$,

We can deduce from [24, Lemma3] that

We easily derive

Since $\tau \to 0$ as $X\to \infty$, by the Prime Number Theorem, we have

Lemma 3.8 ii) can be readily proven by substituting (3.5) and (3.6) into (3.4).

4.   The major arc

In this section, we refine our analysis by dividing the major arc into two distinct regions, $\mathcal{M}{{}_{1}}$ and $\mathcal{M}{{}_{2}}$, which are defined as follows:

4.1.   The region $\mathcal{M}{{}_{1}}$

In this subsection, we establish a lower bound for the integral over the $\mathcal{M}{{}_{1}}$. For $k$ = 1, 2, 3, 4, define

Applying the first derivative estimate for trigonometric integrals (see [25, Lemma 4.2]), we derive the bound

Lemma 4.1. We have

$Proof$. It can be easily demonstrated that

where it is shown that ${{J}_{0}}\gg {{\tau }^{2}}{{X}^{\frac{13}{12}}}$ and ${{J}_{k}} = o({{\tau }^{2}}{{X}^{\frac{13}{12}}})$ for $k$ = 1, 2, 3, 4.

The analysis begins by establishing a lower bound for ${{J}_{0}}$,

Utilizing Eqs (2.3) and (4.1), we deduce that the error term in (4.2) satisfies

For brevity, let ${{y}_{j}} = {{\lambda }_{j}}{{x}_{j}}^{j}$ for $j$ = 1, 2, 3, 4. By changing the order of integration and substituting variables, we note that the domains of integration for ${{y}_{1}}$, ${{y}_{2}}$, ${{y}_{3}}$, and ${{y}_{4}}$ are $\left[{{\lambda }_{j}}\eta X, {{\lambda }_{j}}X \right]$. Moreover, the integral satisfies $\left| \sum\limits_{j = 1}^{4}{{{y}_{j}}-v} \right| < \tau$. Considering $\left| \sum\limits_{j = 1}^{4}{{{y}_{j}}-v} \right| < \frac{\tau }{2}$ and the bounds on ${{y}_{4}}$, we derive the lower bound for the final integral. Therefore

which means that

Thus, combining (4.2)–(4.4), we have

Then we turn to dealing with ${{J}_{1}}$, ${{J}_{2}}$, ${{J}_{3}}$, and ${{J}_{4}}$. According to Euler's summation formula, we have

Using (2.3), we estimate ${{J}_{1}}$ as follows:

Utilizing Cauchy's inequality along with Lemma 3.3 and (4.1), we derive

Similarly, from (4.1) and (4.6), we have

Hence, by combining (4.7)–(4.9), it follows that

Following analogous reasoning for ${{J}_{2}}$ by (2.3), we have

Applying Cauchy's inequality along with Lemmas 3.3 and 3.4, it can be further derived that

Similarly, using the estimates derived from (4.1) and (4.6), along with the results from Lemma 3.4, we have

Therefore, combining (4.11)–(4.13) results in the conclusion that

Similar arguments for ${{J}_{3}}$ and ${{J}_{4}}$, utilizing Lemmas 3.3 and 3.4, we obtain

Hence, by combining (4.5), (4.10), (4.14), and (4.15), the proof of Lemma 4.1 is complete.

4.2.   The region $\mathcal{M}{{}_{2}}$

In this subsection, we derive an upper bound for the integral over the region $\mathcal{M}{{}_{2}}$ to quantify its contribution to the overall integral.

Lemma 4.2. We conclude that

Proof. By applying Cauchy's inequality, along with the results from (2.3) and Lemmas 3.4 and 3.5, we derive the following:

Subsequently, combining the results from Lemmas 4.1 and 4.2, we conclude the integral estimation over the major arc $\mathcal{M}$, to obtain the following lemma.

Lemma 4.3. We have

5.   The trivial arc

In this section, we demonstrate

Utilizing Cauchy's inequality and the trivial bounds of ${{S}_{1}}({{\lambda }_{1}}\alpha)$, ${{S}_{4}}({{\lambda }_{4}}\alpha)$, we obtain

By the periodicity of ${{S}_{2}}({{\lambda }_{2}}\alpha)$, along with (2.3) and Lemma 3.6, we obtain

Similarly, we can conclude that

By combining (5.2)–(5.4), we have

Thus, (5.1) follows directly.

6.   The minor arc and the proof of Theorem 1.2

This section provides a precise estimation of the integral over the minor arc and proves Theorem 1.2. We define $m = \overset{\wedge }{\mathop{m}}\, \cup {{m}^{*}}$, with $\rho = \frac{1}{12}$ and $\tau = {{X}^{-\delta }}$.

Let $\mathcal{E} = \mathcal{E}\left(\mathcal{V}, X, \delta \right)$ denote the set of elements $v$ in $\mathcal{V}$ for which the inequality (1.4) has no solution in the prime variables ${{p}_{j}}$ for $j = 1, 2, 3, 4$. Thus, we have $E(\mathcal{V}, X, \delta) = \left| \mathcal{E}\left(\mathcal{V}, X, \delta \right) \right|$. By selecting an appropriate complex number ${{\vartheta }_{v}}$ such that $\left| {{\vartheta }_{v}} \right| = 1$, we can reformulate the integral as follows:

In this case, we have

By applying Cauchy's inequality, we can bound the expression as follows:

Next, we define $\overset{\wedge }{\mathop{m}}\, = {{m}_{1}}\cup {{m}_{2}}\cup {{m}_{3}}$ and ${{m}^{*}} = m\backslash \overset{\wedge }{\mathop{m}}$, with $X = {{q}^{\frac{90}{49}}}$, where

Lemma 6.1. We have

$Proof$. According to (2.3), we have

where $\mathfrak{L}\left(X \right)$ represents the number of solutions to the inequality

with ${{v}_{1}}, {{v}_{2}}\in \mathcal{E}$ and ${\eta X\le {{p}_{1}}^{2}, {{p}_{2}}^{2}\le X}$.

Since $X$ is sufficiently large and $\tau = {{X}^{-\delta }}$. When ${{v}_{1}} = {{v}_{2}}$, there must exist the case that ${{p}_{1}} = {{p}_{2}}$. Under this condition, we can deduce that

When ${{v}_{1}}\ne {{v}_{2}}$, there exists at most one integer $n$ such that $n\ll X$ and the inequality $\left| {{\lambda }_{2}}n-({{v}_{1}}-{{v}_{2}}) \right| < \tau$ holds. For any integer $n$, the number of solutions to $n = {{p}_{1}}^{2}-{{p}_{2}}^{2}$ is bounded by ${{X}^{\varepsilon }}$. Consequently, we obtain

Combining both cases, we obtain

Thus, Lemma 6.1 is established.

Lemma 6.2. We have

Proof. Utilizing Lemmas 3.6 and 3.8 ii) along with the trivial bounds for ${{S}_{3}}\left({{\lambda }_{3}}\alpha \right)$ and ${{S}_{4}}\left({{\lambda }_{4}}\alpha \right)$, we analyze the integrals over ${{m}_{1}}$, ${{m}_{2}}$, and ${{m}_{3}}$ separately.

For the integral over the interval ${{m}_{1}}$, we obtain

For the integral over the interval ${{m}_{2}}$, we obtain

For the integral over the interval ${{m}_{3}}$, we obtain

By combining these estimates, Lemma 6.2 follows immediately.

Lemma 6.3. We have

$Proof$. Employing Harman's method as outlined in ^[14], we partition the region ${{m}^{*}}$ into disjoint sets $S\left({{Z}_{1}}, {{Z}_{3}}, y \right)$ defined as

where ${{Z}_{1}} = {{X}^{1-\frac{12}{5}\rho +2\varepsilon }}{{2}^{{{t}_{1}}}}$, ${{Z}_{3}} = {{X}^{\frac{1}{3}-\frac{1}{3}\rho +2\varepsilon }}{{2}^{{{t}_{2}}}}$, $y = \xi {{2}^{{{t}_{3}}}}$ for some positive integers ${{t}_{1}}$, ${{t}_{2}}$, and ${{t}_{3}}$.

By invoking Lemmas 3.1 and 3.8 i), we obtain integers ${{a}_{1}}$, ${{q}_{1}}$ and ${{a}_{3}}$, ${{q}_{3}}$ such that $\left({{a}_{1}}, {{q}_{1}} \right) = 1$ and $\left({{a}_{3}}, {{q}_{3}} \right) = 1$ satisfying

Note that ${{a}_{1}}{{a}_{3}}\ne 0$.

We further dissect $S\left({{Z}_{1}}, {{Z}_{3}}, y \right)$ into subsets $S\left({{Z}_{1}}, {{Z}_{3}}, y, {{Q}_{1}}, {{Q}_{3}} \right)$ with $\alpha$ satisfying $\left| \alpha \right|\ge y = \xi {{2}^{{{t}_{3}}}}\ge \xi = {{X}^{-\frac{7}{9}-\varepsilon }}$ and $\left| \frac{{{a}_{i}}}{{{\lambda }_{i}}\alpha } \right|\ll{{q}_{i}}$ for $i = 1, 3$, where

Then, we have

Assuming that $\left| {{a}_{3}}{{q}_{1}} \right|$ takes only $R$ distinct values. By the pigeonhole principle, we have $R\ll \frac{y{{Q}_{1}}{{Q}_{3}}}{q}$. Due to bounds on the divisor function, each value of $\left| {{a}_{3}}{{q}_{1}} \right|$ corresponds to significantly fewer than ${{X}^{\varepsilon }}$ pairs ${{a}_{3}}, {{q}_{1}}$. For fixed ${{a}_{3}}$ and ${{q}_{1}}$, the value of $\left| {{a}_{1}}{{q}_{3}} \right|$ is the integral part of ${{a}_{3}}{{q}_{1}}\frac{{{\lambda }_{1}}}{{{\lambda }_{3}}}$; thus there are significantly fewer than ${{X}^{\varepsilon }}$ pairs ${{a}_{1}}, {{q}_{3}}$. Consequently, by (6.7) and (6.8), the length of $S\left({{Z}_{1}}, {{Z}_{3}}, y, {{Q}_{1}}, {{Q}_{3}} \right)$ is

Evaluating the integral over $S\left({{Z}_{1}}, {{Z}_{3}}, y, {{Q}_{1}}, {{Q}_{3}} \right)$, we obtain

Finally, summing over all possible values of ${{Z}_{1}}, {{Z}_{3}}, y, {{Q}_{1}}, {{Q}_{3}}$, we have

Combining Lemmas 6.2 and 6.3, we arrive at Lemma 6.4.

Lemma 6.4. We have

Proof of Theorem 1.2. We now proceed to prove the first part of Theorem 1.2. Substituting (6.4) and Lemma 6.4 into (6.2), we have

Due to $0 < \delta < \frac{11}{288}$, there is $\tau {{X}^{\frac{13}{12}-\frac{11}{288}+\varepsilon }} = o\left({{\tau }^{2}}{{X}^{\frac{13}{12}}} \right)$, we obtain

Thus, we find

Since ${{{\lambda }_{1}}}/{{{\lambda }_{3}}}\;$ is irrational, there exist infinitely many values of $q$ that can be selected with a sequence ${{X}_{j}}\to \infty$ such that

This completes the proof of the first part of Theorem 1.2.

Next, we prove the second part of Theorem 1.2. By the proof methods from Lemmas 6.2 and 6.3, we observe that replacing $\rho$ with $\chi$ (as defined in Theorem 1.2) is sufficient, resulting in the following conditions:

Substituting and simplifying, we obtain

Given the conditions of the theorem, specifically $0 < \delta < \frac{11}{24}\chi$, we have the asymptotic relation $\tau {{X}^{\frac{13}{12}-\frac{11}{24}\chi +\varepsilon }} = o\left({{\tau }^{2}}{{X}^{\frac{13}{12}}} \right)$. From this, we can deduce that

Thus, the second part of Theorem 1.2 is proved.

7.   Conclusions

In this paper, we prove that, for any $\varepsilon > 0$, the number of $v\in \mathcal{V}$ with $v\le X$ such that the inequality

has no solution in primes ${{p}_{1}}$, ${{p}_{2}}$, ${{p}_{3}}$, ${{p}_{4}}$ that does not exceed $O({{X}^{1-\frac{83}{144}+2\delta +2\varepsilon }})$.

Author contributions

Xinyan Li: Writing-review and editing, writing-original draft, validation, resources, methodology, formal analysis, conceptualization; Wenxu Ge: Writing-review and editing, resources, methodology, supervision, validation, formal analysis. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We express our sincere gratitude to the High-level Talent Research Start-up Project Funding of Henan Academy of Sciences (Grant No. 252019083) for providing us with crucial initial funding for our research. Additionally, we would like to thank the National Natural Science Foundation of China (Grant No. 12071132) and the Joint Fund of Henan Province Science and Technology R$&$D Program (Grant No. 225200810032) for their financial support and generous funding. These funds have played an essential role in the successful completion of our research.

Conflict of interest

The authors declare that they have no conflict of interest.