The integral part of a nonlinear form with a square, a cube and a biquadrate

Wenxu Ge; Weiping Li; Feng Zhao

doi:10.1515/math-2020-0069

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The integral part of a nonlinear form with a square, a cube and a biquadrate

Wenxu Ge , Weiping Li und Feng Zhao

Veröffentlicht/Copyright: 21. November 2020

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Abstract

In this paper, we show that if λ 1 , λ 2 , λ 3 are non-zero real numbers, and at least one of the numbers λ 1 , λ 2 , λ 3 is irrational, then the integer parts of λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 are prime infinitely often for integers n 1 , n 2 , n 3 . This gives an improvement of an earlier result.

Keywords: Diophantine inequalities; primes; Davenport-Heilbronn method

MSC 2010: 11P32; 11P55; 11J25

1 Introduction

Gauss proved that every prime p ≡ 1 ( mod 4 ) can be represented as n 1 2 + n 2 2 using the imaginary quadratic field ℚ ( i ) . In 1998, Friedlander and Iwaniec [1] proved that n 1 2 + n 2 4 are prime infinitely often for integers n 1 , n 2 . Furthermore, one can consider such problem with Diophantine inequalities. But at present there are essential difficulties to consider the problem of Friedlander and Iwaniec in Diophantine inequality, since the sequence is thin. In this paper, we consider a weaker result by adding a cube. We also conjecture that

Conjecture 1

Suppose that λ 1 , λ 2 be non-zero real numbers, not all negative, and that at least one of the numbers λ 1 , λ 2 is irrational. Then the integer parts of λ 1 n 1 2 + λ 2 n 2 2 are prime infinitely often for integers n 1 , n 2 .

Harman [2] proved that there is a number δ > 0 such that, for θ a positive irrational and α an arbitrary real number, the equality

| θ ( n 1 2 + n 2 2 ) − p 1 p 2 + α | < ( n 1 2 + n 2 2 ) − δ

has infinitely many solutions with coprime integers n 1 , n 2 and primes p 1 , p 2 . This implies that the integer parts of θ ( n 1 2 + n 2 2 ) are just two prime factors infinitely often for integers n 1 , n 2 .

In 2015, Yang and Li [3] proved that if λ 1 , λ 2 , λ 3 , λ 4 are non-zero real numbers, at least one of the ratios λ _i/λ _j is irrational, then the integer parts of λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 + λ 4 n 4 5 are prime infinitely often for integers n 1 , n 2 , n 3 , n 4 . Later, Li, Ge and Wang [4] proved that the aforementioned result is still valid for primes p 1 , p 2 , p 3 , p 4 .

In this paper, we establish the following result.

Theorem 1

Let λ 1 , λ 2 , λ 3 be non-zero real numbers. Suppose that at least one of the real numbers λ 1 , λ 2 , λ 3 are irrational. Then the integer parts of λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 are prime infinitely often for integers n 1 , n 2 , n 3 .

Notation

Throughout the paper, the letter η denotes a sufficiently small, fixed positive number. The letter ε denotes an arbitrarily sufficiently small positive real number. The letter p, with or without subscript, denotes a prime number. Constants, both explicit and implicit, in Vinogradov symbols may depend on λ 1 , λ 2 , λ 3 . We write e ( x ) = exp ( 2 π i x ) .

2 Outline of the method

We follow the modification of the Hardy-Littlewood method first stated by Davenport and Heilbronn. We can assume that λ 1 , λ 2 , λ 3 are not all negative if necessary, n 2 3 is replaced by ( − n 2 ) 3 . Now let X be some (large) positive quantity, L = log X . We define

(2.1) K ( α ) = sin 1 2 π α π α 2 , A ( x ) = ∫ ℝ K ( α ) e ( α x ) d α .

Then, by [5], it is easy to show that

(2.2) K ( α ) ≪ min ( 1 , | α | − 2 ) , A ( x ) = max 0 , 1 2 − | x | .

Let

A = { 1 ≤ n ≤ X 1 / 3 : p | n ⇒ p ≤ X δ } ,

where δ is a sufficiently small, fixed positive number. Obviously, we have

| A | ≫ X 1 / 3 − δ .

We write for j = 1 , 2 ,

(2.3) f ( α ) = ∑ η X < p ≤ X ( log p ) e ( p α ) ; g 3 ∗ ( α ) = ∑ η X < n 3 ≤ X n ∈ A e ( n 3 α ) ;

(2.4) g 2 ( α ) = ∑ η X < n 2 ≤ X e ( n 2 α ) ; g 4 ( α ) = ∑ η X < n 4 ≤ X e ( n 4 α ) ;

I j ( α ) = ∫ ( η X ) 1 / j X 1 / j e ( α x j ) d x ,

where η is a sufficiently small, fixed positive number.

We define further

(2.5) F ( α ) ≔ f ( − α ) g 2 ( λ 1 α ) g 3 ∗ ( λ 2 α ) g 4 ( λ 3 α ) .

For any measurable subset X of ℝ , we define

(2.6) J ( X ) ≔ ∫ X F ( α ) K ( α ) e − 1 2 α d α .

Then by (2.1), (2.2), (2.3) and (2.4), we have

(2.7) J ( ℝ ) = ∑ η X < p , n 1 2 , n 2 3 , n 3 4 ≤ X n 2 ∈ A ( log p ) A λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 − p − 1 2 ≤ 1 2 L ∑ η X < p , n 1 2 , n 2 3 , n 3 4 ≤ X n 2 ∈ A λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 − p − 1 2 < 1 2 1 ≤ 1 2 L ∑ η X < p , n 1 2 , n 2 3 , n 3 4 ≤ X λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 − p − 1 2 < 1 2 1 ≤ 1 2 L N ( X ) ,

where N ( X ) denote the number of solutions of the inequality

(2.8) λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 − p − 1 2 < 1 2

in prime η X < p ≤ X and positive integers η X < n 1 2 , n 2 3 , n 3 4 ≤ X .

To estimate the left integral in (2.7), we divide the real line into three parts: the major arc M , the minor arc m and the trivial arc t , which are defined by

M = { α : | α | ≤ ϕ } , m = { α : ϕ < | α | ≤ ξ } , t = { α : | α | > ξ } ,

where ϕ = X − 1 / 2 − ε , ξ = X 2 δ .

3 Preliminary lemmas

Lemma 1

We have

(3.1) ∫ 0 1 | f ( α ) | 2 d α ≪ X 1 + ε , ∫ − ∞ + ∞ | f ( α ) | 2 K ( α ) d α ≪ X 1 + ε ;

(3.2) ∫ 0 1 | g 2 ( α ) 4 | K ( α ) d α ≪ X 1 + ε , ∫ − ∞ + ∞ | g 2 ( α ) 4 | K ( α ) d α ≪ X 1 + ε ;

(3.3) ∫ 0 1 | g 2 ( α ) 2 g 4 ( α ) 4 | d α ≪ X 1 + ε , ∫ − ∞ + ∞ | g 2 ( α ) 2 g 4 ( α ) 4 | K ( α ) d α ≪ X 1 + ε .

Proof

Estimates (3.1) and (3.2) follow from Hua’s Lemma (see Lemma 2.5 of [6]). Estimate (3.3) follows from Lemma 4.2 of [7].□

Lemma 2

[8, Theorem 1.4] Suppose that δ > 0 and k ≥ 7.5906 . Then one has

∫ 0 1 | g 3 ∗ ( α ) | k d α ≪ X k / 3 − 1 , ∫ − ∞ + ∞ | g 3 ∗ ( α ) | k K ( α ) d α ≪ X k / 3 − 1 .

Lemma 3

([6], see also [9, Corollary 3.2]) Suppose that X ≥ Z ≥ X 4 / 5 + ε and that | f ( α ) | > Z . Then there are coprime integers a , q satisfying

1 ≤ q ≪ X 2 + ε Z − 2 , | q α − a | ≪ X 1 + ε Z − 2 .

Lemma 4

[10] Suppose that X 1 / 2 ≥ Z ≥ X 1 / 4 + ε and that | g 2 ( α ) | > Z . Then there are coprime integers a , q satisfying

1 ≤ q ≪ X 1 + ε Z − 2 , | q α − a | ≪ X ε Z − 2 .

Lemma 5

[11, Lemma 2.2] Suppose that α is a real number and suppose that a ∈ Z and q ∈ ℕ satisfy ( a , q ) = 1 , q ≤ X 2 / 3 and | q α − a | < X − 2 / 3 . Then

g 3 ∗ ( α ) ≪ X 3 / 10 + ε + X 1 / 3 + ε q 1 / 6 ( 1 + X | α − a / q | ) 1 / 3 .

Corollary 1

Suppose that X 1 / 3 ≥ Z ≥ X 3 / 10 + ε and that | g 3 ∗ ( α ) | > Z . Then there are coprime integers a , q satisfying

1 ≤ q ≪ X 2 + ε Z − 6 , | q α − a | ≪ X 1 + ε Z − 6 .

Proof

By Lemma 5, we have

Z < | g 3 ∗ ( α ) | ≪ X 3 / 10 + ε / 6 + X 1 / 3 + ε / 6 q 1 / 6 ( 1 + X | α − a / q | ) 1 / 3 .

Thus, we have

q 1 / 6 ≪ X 1 / 3 + ε / 6 Z − 1 , q 1 / 6 X 1 / 3 | α − a / q | 1 / 3 ≪ X 1 / 3 + ε / 6 Z − 1 .

Then

q ≪ X 2 + ε Z − 6 , | q α − a | ≪ q 1 / 2 X ε / 2 Z − 3 ≪ X 1 + ε Z − 6 .

□

4 The major arc

Lemma 6

For ε > 0 and j = 1 , 2 , we have

(4.1) ∫ M | f ( α ) − I 1 ( α ) | 2 d α ≪ X L − 2 , ∫ M | I j ( α ) | 2 d α ≪ X 2 / j − 1 ,

(4.2) ∫ M | g 2 ( λ 1 α ) − I 2 ( λ 1 α ) | 2 d α ≪ X − ε , ∫ M | g 2 ( λ 1 α ) | 2 d α ≪ 1 .

Proof

The first estimate of (4.1) follows from Lemma 2 of [12] and that

I 1 ( α ) − ∑ η X < n ≤ X e ( α n ) ≪ 1 + | α | X .

The second estimate of (4.1) follows from that

(4.3) | I j ( α ) | ≪ X 1 / j min ( 1 , | α | − 1 X − 1 ) .

By Theorem 4.1 of [6], we have | g 2 ( α ) − I 2 ( α ) | ≪ ( 1 + | α | X ) 1 / 2 . Thus,

∫ M | g 2 ( α ) − I 2 ( α ) | 2 d α ≪ X ϕ 2 ≪ X − ε

and

∫ M | g 2 ( α ) | 2 d α ≪ ∫ M | g 2 ( α ) − I 2 ( α ) | 2 d α + ∫ M | I 2 ( α ) | 2 d α ≪ 1 .

□

Lemma 7

Let

G ( α ) = I 1 ( − α ) I 2 ( λ 1 α ) g 3 ∗ ( λ 2 α ) g 4 ( λ 3 α ) .

Then we have

(4.4) ∫ M | F ( α ) − G ( α ) | K ( α ) d α = o ( X 3 / 4 | A | ) .

Proof

By (2.2), (2.3), (2.4) and Lemma 6, we have

∫ M | F ( α ) − G ( α ) | K ( α ) d α ≪ X 1 / 4 | A | ∫ M | f ( − α ) g 2 ( λ 1 α ) − I 1 ( − α ) I 2 ( λ 1 α ) | d α ≪ X 1 / 4 | A | ∫ M | f ( − α ) g 2 ( λ 1 α ) − I 1 ( − α ) g 2 ( λ 1 α ) | d α + X 1 / 4 | A | ∫ M | I 1 ( α ) g 2 ( λ 1 α ) − I 1 ( − α ) I 2 ( λ 1 α ) | d α ≪ X 1 / 4 | A | ∫ M | f ( α ) − I 1 ( α ) | 2 d α 1 / 2 ∫ M | g 2 ( λ 1 α ) | 2 d α 1 / 2 + X 1 / 4 | A | ∫ M | g 2 ( λ 1 α ) − I 2 ( λ 1 α ) | 2 d α 1 / 2 ∫ M | I 1 ( α ) | 2 d α 1 / 2 ≪ X 3 / 4 | A | L − 1 .

□

Lemma 8

We have

(4.5) J ( M ) ≫ X 3 / 4 | A | .

Proof

We only need to prove that

(4.6) ∫ M G ( α ) K ( α ) e − 1 2 α d α ≫ X 3 / 4 | A | .

Then Lemma 8 immediately follows from Lemma 7. By (4.3), we have

(4.7) ∫ M G ( α ) K ( α ) e − 1 2 α d α = ∫ − ∞ + ∞ G ( α ) K ( α ) e − 1 2 α d α + O ∫ | α | > ϕ | G ( α ) | K ( α ) d α = ∫ − ∞ + ∞ G ( α ) K ( α ) e − 1 2 α d α + O X 1 / 4 | A | ∫ | α | > ϕ | I 1 ( α ) I 2 ( λ 1 α ) | d α

= ∫ − ∞ + ∞ G ( α ) K ( α ) e − 1 2 α d α + O X 1 / 4 | A | X − 1 / 2 ∫ | α | > ϕ | α | − 2 d α = ∫ − ∞ + ∞ G ( α ) K ( α ) e − 1 2 α d α + O ( X 1 / 4 + ε | A | ) .

Then claim (4.6) follows by showing that

∫ − ∞ + ∞ G ( α ) K ( α ) e − 1 2 α d α ≫ X 3 / 4 | A | ,

which follows as Lemma 10 of [5].□

5 The trivial arc

By Lemmas 1, 2 and (2.2), we have

J ( t ) ≪ ∫ | α | > ξ | f ( α ) g 2 ( λ 1 α ) g 3 ∗ ( λ 2 α ) g 4 ( λ 3 α ) | K ( α ) d α ≪ ∫ | α | > ξ | f ( α ) | 2 α − 2 d α 1 / 2 ∫ | α | > ξ | g 2 ( λ 1 α ) 2 g 4 ( λ 3 α ) 4 | α − 2 d α 1 / 4 × ∫ | α | > ξ | g 2 ( λ 1 α ) | 4 α − 2 d α 1 / 8 ∫ | α | > ξ | g 3 ∗ ( λ 2 α ) | 8 α − 2 d α 1 / 8 ≪ ∑ k ≥ [ ξ ] 1 k 2 ∫ 0 1 | f ( α ) | 2 d α 1 / 2 ∑ k ≥ [ ξ ] 1 k 2 ∫ 0 1 | g 2 ( α ) 2 g 4 ( α ) 4 | d α 1 / 4 × ∑ k ≥ [ ξ ] 1 k 2 ∫ 0 1 | g 2 ( α ) | 4 d α 1 / 8 ∑ k ≥ [ ξ ] 1 k 2 ∫ 0 1 | g 3 ∗ ( α ) | 8 d α 1 / 8 ≪ ξ − 1 ∫ 0 1 | f ( α ) | 2 d α 1 / 2 ∫ 0 1 | g 2 ( α ) 2 g 4 ( α ) 4 | d α 1 / 4 × ∫ 0 1 | g 2 ( α ) | 4 d α 1 / 8 ∫ 0 1 | g 3 ∗ ( α ) | 8 d α 1 / 8 ≪ ξ − 1 X 1 2 + 1 4 + 1 8 + 5 24 + ε ≪ X 13 12 − 2 δ + ε .

Thus, we have

(5.1) J ( t ) = o ( X 3 / 4 | A | ) ,

since | A | ≫ X 1 / 3 − δ .

6 The proof of Theorem 1 for the cases λ 1 or λ 3 irrational

The cases λ 1 irrational and λ 3 irrational are similar, and we only list the proof of the case λ 1 irrational here. First, we divide the minor arc m into three parts. Let 0 < Δ < 1 / 10 and m 3 = m \ ( m 1 ∪ m 2 ) , where

(6.1) m 1 = { α ∈ m : | f ( α ) | ≪ X 1 − Δ + ε } , m 2 = { α ∈ m : | g 2 ( α ) | ≪ X 1 2 ( 1 − Δ ) + ε } .

For convenience, we take s = 7.5906 . Then by Lemmas 1 and 2, we have

J ( m 1 ) ≪ max α ∈ m 1 | f ( α ) | 2 s − 1 4 ∫ − ∞ + ∞ | f ( α ) | 2 K ( α ) d α 5 8 − 1 s ∫ − ∞ + ∞ | g 3 ∗ ( α ) | s K ( α ) d α 1 / s × ∫ − ∞ + ∞ | g 2 ( α ) 2 g 4 ( α ) 4 | K ( α ) d α 1 4 ∫ − ∞ + ∞ | g 2 ( α ) 4 | K ( α ) d α 1 8 ≪ X 3 / 4 X 1 / 3 − Δ 2 s − 1 4 + ε

and

J ( m 2 ) ≪ max α ∈ m 2 | g 2 ( α ) | 4 s − 1 2 ∫ − ∞ + ∞ | f ( α ) | 2 K ( α ) d α 1 2 ∫ − ∞ + ∞ | g 3 ∗ ( α ) | s K ( α ) d α 1 / s × ∫ − ∞ + ∞ | g 2 ( α ) 2 g 4 ( α ) 4 | K ( α ) d α 1 4 ∫ − ∞ + ∞ | g 2 ( α ) 4 | K ( α ) d α 1 4 − 1 s ≪ X 3 / 4 X 1 / 3 − Δ 2 s − 1 4 + ε .

Since δ is a sufficiently small positive number, we take 0 < δ < Δ 2 s − 1 4 . Thus, we have

(6.2) J ( m 1 ∪ m 2 ) ≪ X 3 / 4 X 1 / 3 − Δ 2 s − 1 4 + ε ≪ X 3 / 4 X 1 / 3 − δ = o ( X 3 / 4 | A | ) .

It remains to discuss the set m 3 . We use the method of Harman [13]. We divide m 3 into disjoint sets such that for α ∈ ℬ ( Z 1 , Z 2 , y ) , we have

Z 1 ≤ | f ( α ) | < 2 Z 1 , Z 2 ≤ | g 2 ( λ 1 α ) | < 2 Z 2 , y ≤ | α | < 2 y ,

where Z 1 = X 1 − Δ + ε 2 t 1 , Z 2 = X 1 / 2 − Δ / 2 + ε 2 t 2 , y = ϕ 2 u for some positive integers t 1 , t 2 , u . Thus, by Lemmas 3 and 4, there exist two pairs of coprime integers ( a 1 , q 1 ) , ( a 2 , q 2 ) with a 1 a 2 ≠ 0 and

1 ≤ q 1 ≪ X 2 + ε Z 1 − 2 , | q 1 α − a 1 | ≪ X 1 + ε Z 1 − 2 ; 1 ≤ q 2 ≪ X 1 + ε Z 2 − 2 , | q 2 λ 1 α − a 2 | ≪ X ε Z 2 − 2 .

Let ℬ = ℬ ( Z 1 , Z 2 , y , Q 1 , Q 2 ) be the subset of ℬ ( Z 1 , Z 2 , y ) for which q j ∼ Q j . Then, by a familiar argument (see p. 147 of [6] for example),

| a 2 q 1 − λ 1 a 1 q 2 | = a 2 ( q 1 α − a 1 ) + a 1 ( a 2 − q 2 λ 1 α ) α ≪ Q 2 X 1 + ε Z 1 − 2 + Q 1 X ε Z 2 − 2 ≪ X 2 + 2 ε Z 1 − 2 Z 2 − 2 ≪ X − 1 + 3 Δ − 2 ε .

Also,

| a 1 q 2 | ≪ y Q 1 Q 2 .

Since λ 1 is irrational, we let a/q be a convergent to λ 1 and put X = q 1 1 − 3 Δ . Thus, we have

(6.3) ∥ λ 1 a 1 q 2 ∥ ≤ 1 4 q , q 2 ∼ Q 2 , a 1 ≈ y Q 1 ,

since X is sufficiently large. Then by the pigeon-hole principle and the Legendre law of best approximation for continued fractions, the aforementioned inequality (6.3) has ≪ y Q 1 Q 2 q − 1 solutions of | a 1 q 2 | . Clearly, each value of | a 1 q 2 | corresponds to ≪ X ε values of a 1 , a 2 , q 1 , q 2 by the well-known bound on the divisor function. Hence, we conclude that

(6.4) μ ( ℬ ) ≪ X ε y Q 1 Q 2 q min ( X 1 + ε Z 1 − 2 Q 1 − 1 , X ε Z 2 − 2 Q 2 − 1 ) ≪ X ε y Q 1 Q 2 q X 1 / 2 + ε Z 1 Z 2 Q 1 1 / 2 Q 2 1 / 2 ≪ X 1 / 2 + 2 ε y Q 1 1 / 2 Q 2 1 / 2 q Z 1 Z 2 ≪ X 2 + 2 ε y q Z 1 2 Z 2 2 ,

where μ ( ℬ ) is the Lebesgue measure of ℬ . Thus, we have

∫ ℬ | f ( α ) g 2 ( α ) g 3 ∗ ( α ) g 4 ( α ) | K ( α ) d α ≪ Z 1 Z 2 | A | X 1 / 4 min ( 1 , | y | − 2 ) X 2 + 2 ε y q Z 1 2 Z 2 2 ≪ | A | X 1 / 4 X 2 + 2 ε y q Z 1 Z 2 ≪ | A | X 9 Δ / 2 − 1 / 4 + ε .

Summing over all possible values of Z 1 , Z 2 , y , Q 1 , Q 2 , we conclude that

(6.5) J ( m 3 ) ≪ | A | X 9 Δ / 2 − 1 / 4 + 6 ε = o ( X 3 / 4 | A | ) .

From (2.7), (4.5), (5.1), (6.2) and (6.5), we have

N ( X ) ≫ X 3 / 4 | A | L − 1 .

Since λ 1 is irrational, there are infinitely many pairs of integers q , a . So X = q 1 1 − 3 Δ → ∞ , as q → ∞ , and this implies that the integer parts of λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 are prime infinitely often for integers n 1 , n 2 , n 3 . This completes the proof of Theorem 1 for the case λ 1 irrational.

7 The proof of Theorem 1 for the case λ 2 irrational

Let m 3 ′ = m \ ( m 1 ∪ m 2 ′ ) , where m 1 is defined in (6.1) and

m 2 ′ = α ∈ m : | g 3 ∗ ( α ) | ≪ X 1 3 ( 1 − Δ ) + ε .

Then by Lemmas 1 and 2, we have

J ( m 2 ′ ) ≪ max α ∈ m 2 g 3 ∗ ( α ) 1 − s 8 ∫ − ∞ + ∞ | f ( α ) | 2 K ( α ) d α 1 2 ∫ − ∞ + ∞ | g 3 ∗ ( α ) | s K ( α ) d α 1 8 × ∫ − ∞ + ∞ | g 2 ( α ) 2 g 4 ( α ) 4 | K ( α ) d α 1 4 ∫ − ∞ + ∞ | g 2 ( α ) 4 | K ( α ) d α 1 8 ≪ X 3 / 4 X 1 / 3 − Δ 1 3 − s 24 + ε = o ( X 3 / 4 | A | ) ,

since 0 < δ < Δ ( 2 s − 1 4 ) < Δ ( 1 3 − s 24 ) .

It remains to discuss the set m 3 ′ . We divide m 3 ′ into disjoint sets such that for α ∈ ℬ ′ ( Z 1 , Z 2 , y ) , we have

Z 1 ≤ | f ( α ) | < 2 Z 1 , Z 2 ≤ | g 3 ∗ ( λ 1 α ) | < 2 Z 2 , y ≤ | α | < 2 y ,

where Z 1 = X 1 − Δ + ε 2 t 1 , Z 2 = X 1 3 ( 1 − Δ ) + ε 2 t 2 , y = ϕ 2 u for some positive integers t 1 , t 2 , u . Thus, by Lemma 3 and Corollary 1, there exist two pairs of coprime integers ( a 1 , q 1 ) , ( a 2 , q 2 ) with a 1 a 2 ≠ 0 and

1 ≤ q 1 ≪ X 2 + ε Z 1 − 2 , | q 1 α − a 1 | ≪ X 1 + ε Z 1 − 2 , 1 ≤ q 2 ≪ X 2 + ε Z 2 − 6 , | q 2 λ 2 α − a 2 | ≪ X 1 + ε Z 2 − 6 .

Let ℬ ′ = ℬ ′ ( Z 1 , Z 2 , y , Q 1 , Q 2 ) be the subset of ℬ ′ ( Z 1 , Z 2 , y ) for which q j ∼ Q j . Then

| a 2 q 1 − λ 2 a 1 q 2 | ≪ X 3 + 2 ε Z 1 − 2 Z 2 − 6 ≪ X − 1 + 4 Δ − 6 ε .

Also,

| a 1 q 2 | ≪ y Q 1 Q 2 .

Let a ′ / q ′ be a convergent to the irrational number λ 3 and put X = q ′ 1 1 − 4 Δ . Thus, we have

(7.1) μ ( ℬ ′ ) ≪ X ε y Q 1 Q 2 q ′ min ( X 1 + ε Z 1 − 2 Q 1 − 1 , X 1 + ε Z 2 − 6 Q 2 − 1 ) ≪ X 3 + 3 ε y q ′ Z 1 2 Z 2 6 .

Then

∫ ℬ ′ | f ( α ) g 2 ( α ) g 3 ∗ ( α ) g 4 ( α ) | K ( α ) d α ≪ Z 1 Z 2 X 1 / 2 + 1 / 4 min ( 1 , | y | − 2 ) X 3 + 3 ε y q ′ Z 1 2 Z 2 6 ≪ X 1 / 12 + 20 Δ / 3 + ε .

Summing over all possible values of Z 1 , Z 2 , y , Q 1 , Q 2 , we conclude that

J ( m 3 ′ ) ≪ X 1 / 12 + 20 Δ / 3 + 6 ε .

Then Theorem 1 for the case λ 2 irrational follows by repeating the works in Section 6.

Acknowledgments

The authors thank the referees for their time and comments. The research has been supported by the National Natural Science Foundation of China (Grant No. 11871193 and 11471112) and the Key Research Project of Henan Province Higher Education (No. 17A110009).

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Received: 2019-12-10

Revised: 2020-07-11

Accepted: 2020-07-23

Published Online: 2020-11-21

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