Abstract
Let be an integer with and be any real number. Suppose that are nonzero real numbers, not all of them have the same sign, and is irrational. It is proved that the inequality has infinitely many solutions in prime variables , where , and . This gives an improvement of the recent results.
1. Introduction
The determination of the minimal such that the Diophantine equationis solvable in positive integers , for all sufficiently large integers is an interesting problem in additive number theory. In 1951, Roth [1] proved that is acceptable. This result was subsequently improved by Thanigasalam et al. [2–4], Vaughan and Vaughan [5, 6], Brüdern and Brüdern [7, 8] and Ford and Ford [9, 10]. The best currently known result is due to Ford [10], with . Schwarz [11] suggested to analyze the related Diophantine inequality. The first result was obtained by Brüdern [12], who showed that the values ofat integer points are dense on the real line provided that are nonzero real numbers and is irrational. Thanks to a pruning technique, Brüdern [13] proved that the values taken byat integer points are dense on the real line if are nonzero real numbers and at least one of the ratios is irrational.
Suppose that are prime variables, is a sufficiently large integer, and is even. In 1969, Vaughan proved in his doctoral thesis that (1) is solvable if . Later, Vaughan [5] improved upon his own result by taking in place of . By calculating, the exponential density more accurately, Shan [14] showed that is acceptable. In addition, Prachar [15] established that each sufficiently large odd integer can be represented aswhere are prime numbers. As a corollary of [16] in Theorem 1, Ren and Tsang obtained the same result as Prachar. It is of some interest to consider the analogous form for Diophantine inequalities. Let be nonzero real numbers, not all of them have the same sign and as irrational. In 2016, Ge and Li [17] proved that, for any given real numbers and , there exist infinitely many solutions in prime numbers to the inequality
Let be an integer. The first author [18] investigated the solvability of more general Diophantine inequalityand proved that (6) has infinitely many solutions in prime variables for and with . Subsequently, Liu [19] obtained . In [20], the first author and Qu showed that is acceptable. Very recently, this result was improved by Zhu [21], who obtained . In [22], Gao and Liu gave an improvement ([18] in Theorem 1.2) in case , and they proved particularly.
The main purpose of this paper is to sharpen the above results in case . We obtain the following theorem.
Theorem 1. Let be an integer with and be any given real number. Suppose that are nonzero real numbers, not all of them are same sign, and is irrational. Then, inequality (6) has infinitely many solutions in prime variables , where , and .
The improvement derives not only from the use of the function constructed by Harman and Kumchev (see Section 8 in [23] and Section 5 in [24], for details) but also from some ingredients in [21]. It is worth remarking that Ge et al. [25] obtained , if the condition “ is irrational” in Theorem 1 is replaced by “ is irrational and and are rational.”
Notation. Throughout the paper, and are arbitrarily small, fixed positive real numbers. Any statement in which occurs holds for each positive . The implicit constants in -term, - and -symbols depend at most on and . The letter , with or without subscript, is reserved for a prime number. By , we mean that and . For simplicity, we write and .
2. Preliminaries
We apply the Davenport–Heilbronn circle method (see [26] and Chapter 11 in [27]) to prove Theorem 1. Since is irrational, there are infinitely many convergents to its continued fraction. Let be any denominator of a convergent to . As in [20], let run through the sequences:
We setwhere the function is defined by 5.2 in [24]. According to [24], is a nontrivial lower bound for the characteristic function of the set of primes in , and it satisfies
For further properties of , see Lemma 1 and (4.2)–(4.4) in [24]. Let
By the prime number theorem, it is easy to show that . For any fixed , set for and . Clearly, we have
A straightforward application of the Cauchy integral formula gives
Identity (12) is also a corollary of Lemma 4 in [26]. For , put
We writefor any measurable subset of . It follows from (9) and (12) thatwhere denotes the number of solutions of the inequalitywith , and for . In what follows, we takeactually. We now divide the real line into three disjoint parts:where . These sets are called the major arc, the minor arcs, and the trivial regions, respectively.
In the following sections, we shall prove that the dominant contribution to is from the major arc, and the contribution from the minor arcs and the trivial region can be neglected.
3. The Major Arc
Our first goal is to show that
The proof of (19) is quite similar to that given in Section 3 in [20]. For completeness of exposition, we briefly present the proof procedure below.
Let
Then, we have and
By a similar argument as that in pp. 1656–1657 in [20], we can obtain
To estimate the integrals and , we need the following two lemmas.
Lemma 1. Let be an integer. Then, for nonzero real number and any , we have
Proof. It follows from Theorem 1 in [28].
Lemma 2. For , suppose thatThen, we have
Proof. See Lemma 3.7 in [20].
When , it follows from (23) that
Combining this with the Cauchy–Schwarz inequality and Lemma 2 giveswhere (11) is used.
When , (23) implies
Proceeding as in the proof of (27), we have
This with (27), (22), and (21) yields (19).
4. The Minor Arcs
The next thing to do in the proof is to establish that
This work forms the bulk of the present paper. We subdivide into four disjoint parts: , where
Therefore,
To prove (30), it suffices to show that holds for .
We apply Hölder’s inequality and Lemma 2 to estimate . When , we have
If , then
In case , we obtain
It follows from (33)–(35) and (17) that
In order to establish an upper bound for as small as possible, we need the following lemma.
Lemma 3. (Lemma 3.4 in [21]). LetThen, we have
For , by the Cauchy–Schwarz inequality, Lemmas 2 and 3, we obtainwhere the trivial upper bound is used. It is easily derived from (17) that
The upper bound estimation of plays a crucial role in the proof. The parameter , which is given by (17), is determined in this step. When , by Hölder’s inequality and Lemma 2, we have
In the case of , we obtain
If , we deduce that
Inequalities (41)–(43) and (17) together give
In the remainder of this section, we shall be trying to estimate . By a familiar dyadic dissection argument, we divide into at most disjoint sets . For , we havewhere , and for some nonnegative integers . For the sake of convenience, we take the notation as a shortcut for , and let stand for the Lebesgue measure of .
Lemma 4. (Lemma 4.3 in [20]). We have
When , it follows from (11) and Hölder’s inequality thatwhere Lemmas 2 and 3 are used.
When , by the similar argument as in the proof of (47), we obtain
When , we have
Thanks to (17) and (47)–(49), we are led to the conclusion that
This together with (36), (40), (44), and (32) gives (30).
5. The Trivial Regions
Finally, it only remains to treat . Suppose that and are positive integers with . For any and nonzero real , we have (see (5.1) and (5.2) in [20])
It follows from Hölder’s inequality that
Recalling that and inserting this expression into (52) yields
6. Completion of the Proof
We are now in a position to get the desired conclusion. It should be noted that
From this and (19), (30), and (53), we infer that . Hence, by (15),
This implies inequality (16) has solutions in quintuples of primes with , and , for . Notice that is irrational, is any denominator of a convergent to and . By substituting (17) into (55), we deduce that as . In view ofand (17), we obtain the required range of in Theorem 1. This completes the proof of Theorem 1.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of the first author was supported by the Natural Science Basic Research Plan in Shaanxi Province of China (no. 2019JM–337), China Scholarship Council (CSC) Scholarship Program (no. 202008615008), and Scientific Research Foundation of Xi’an Polytechnic University (no. BS1508).