On a Diophantine inequality involving a prime and an almost-prime
aa r X i v : . [ m a t h . N T ] M a y ON A DIOPHANTINE INEQUALITY INVOLVING A PRIMEAND AN ALMOST-PRIME
LIYANG YANG
Abstract.
We prove that there are infinitely many solutions of | λ + λ p + λ P r | < p − τ , where r = 3 , τ = , and λ is an arbitrary real number and λ , λ ∈ R with λ = 0 and > λ λ not in Q . This improves a result by Harman. Moreover, weshow that one can require the prime p to be of the form ⌊ n c ⌋ for some positiveinteger n , i.e. p is a Piatetski-Shapiro prime, with r = 13 and τ = ρ ( c ) , aconstant explicitly determined by c supported in (cid:0) , (cid:3) . Contents
1. Introduction 12. Notation and outline of the method 22.1. Notation 22.2. The weighted sieve 33. Some auxiliary lemmas 54. Estimates for exponential sums I 65. Sieve estimates 96. Proof of Theorem 1 187. Estimates for exponential sums II 208. Proof of Theorem 2 22References 231.
Introduction
In Diophantine Approximation, a classical theorem of Kronecker ([4], Theorem440) indicates that there are infinitely many solutions in positive integers n , n of | λ + λ n + λ n | < (cid:18) max (cid:26) n λ , n λ (cid:27)(cid:19) − , where λ λ is irrational and λ is an arbitrary real number.The case where n and n are both primes is of great interest and remains opento date ([12], [13]). The first approximation in this direction has been given byVaughan [14] who proved that there are infinitely many solutions of | λ + λ p + λ P | < p − / , where and henceforth in this paper the letter p denotes a prime and P r a numberwith at most r prime factors. Harman [6] proved that there are infinitely manysolutions of(1) | λ + λ p + λ P | < p − τ , Date : November 13, 2018. with τ = .In this paper, we will improve Harman’s result by showing that in (1) one canactually take τ = . One of the main results of this paper will be the following. Theorem 1.
For λ , λ , λ ∈ R with λ λ both negative and irrational, there areinfinitely many solutions of | λ + λ p + λ P | < p − . Moreover, recall that in [7] Heath-Brown proved Pjatecki- ˇ Sapiro prime numbertheorem, i.e. π c ( x ) := X n ≤ x ⌊ n c ⌋ is a prime c − Li ( x ) + O (cid:16) xe − δ √ log x (cid:17) , where c is a real number satisfying that < c < = 1 . ... , and δ = δ ( c ) > . Thus we can naturally ask, what will happen if we replace the prime numbertheorem in the main term by Pjatecki- ˇ Sapiro prime number theorem? Can werequire the prime p in Theorem 1 to be a Pjatecki- ˇ Sapiro prime?The answer is positive, although at cost of increasing the number of factors ofthe corresponding almost-prime, and we will give a concrete describe about it asfollows.
Theorem 2.
For c ∈ (cid:0) , (cid:3) , λ , λ , λ ∈ R with λ λ both negative and irra-tional, there are infinitely many solutions of | λ + λ ˜ p + λ P | < ˜ p − ρ ( c ) , where ˜ p is a prime of the form ⌊ n c ⌋ for some positive integer n and ρ ( c ) := 1 + 9( c − − − c − . . Remark.
We can take ρ ( c ) = , when c = 1 + 2 × − . Acknowledgements. abc2.
Notation and outline of the method
Notation.
We shall use η and ε for arbitrary small positive numbers (espe-cially we require ε ≤ η ≤ − ); and sometimes they may be slightly different incontext just for simplicity.We write ⌊ x ⌋ for the largest integer not exceeding x . We write k x k for thedistance from x to a nearest integer and ⌈ x ⌋ for the nearest integer to x when k x k 6 = . Clearly we may assume that λ > and λ = − . Let a ′ q be a convergenceto the continued fraction for λ and assume q to be quite large in terms of λ , λ and λ − ; let X be a large number such that q ≍ X + ρ + η . Trivially, one can write λ = bq + γ with | γ | < q .As in [6], we assume that q is so large that min { a ′ q , qa ′ } > X − ρ and a ′ X + b ′ The weighted sieve. Essentially, to prove Theorem 1, if we use the samemethod as in [6] but with a parameterized weight to optimize the result, we willobtain that τ = is admissible as mentioned in Section 6. However, one canexpect to obtain a better result by using Buchstab’s sifting weights in [10] ratherthan Richert’s weight w p := 1 − u log p log X , together with Selberg’s trick, as in [8]. Wewill show in Theorem 14 that some terms in the resulting sums can be estimatedmore efficiently by using a 2-dimensional sieve, rather than using the linear sieveonly. The 2-dimensional sieve helps us sieve primes in a much larger range, whichwill give a better result. Moreover, combining with Chen’s idea, i.e., the so-calledSwitching Principle, as in [6], we can thus improve Harman’s result. The last stepis to work out the restrictions of those parameters both from main terms and errorterms explicitly, and then figure out the optimal results from them, which can bedone by Mathematica 9 .We will put the proof Theorem 2 in the last section, as it’s somewhat similarto that of Theorem 1. For instance, the exponential sums appearing in the errorterms can actually be divided into two parts roughly, one of which can actually behandled by results in Section 4. Nevertheless, we need a lemma to estimate theother part because it is an exponential sum of analytic type. All these will be donein Section 7.Also, we will cover a slight gap of [6] in Section 4. Remark. Selberg’s trick can often help us slightly expand the range of sifting, e.g.see [9], where the sifting set is naturally multiplicative by the Chinese remindertheorem, and thus is easier to handle. However, the sifting set here has no multi-plicative structure, so we have to use other tricks to conquer.As it points out in [6] it suffices to show that the number of solutions of (cid:12)(cid:12)(cid:12)(cid:12) b ′ q + pa ′ q − P (cid:12)(cid:12)(cid:12)(cid:12) < X − ρ tends to infinity with X . Here p < X , P < a ′ X + b ′ q . Hence, we will work with theset A := (cid:26)(cid:24) b ′ + pa ′ q (cid:23) : p X, (cid:13)(cid:13)(cid:13)(cid:13) b ′ + pa ′ q (cid:13)(cid:13)(cid:13)(cid:13) < ξ (cid:27) . Here we list all notation used in the sieve method: H r := { n ∈ H : r | n } , for any finite set of positive integers H ; N ( β ) := ( p p p p : X β p < X β , p p (cid:18) a ′ X + b ′ qp (cid:19) ,p p (cid:18) a ′ X + b ′ qp p (cid:19) , X α p a ′ X + b ′ qp p p ) ; A ( β ) ∗ := (cid:26) n : n X, (cid:13)(cid:13)(cid:13)(cid:13) b ′ + na ′ q (cid:13)(cid:13)(cid:13)(cid:13) < ξ , (cid:24) b ′ + na ′ q (cid:23) ∈ N ( β ) (cid:27) ; P r := { n ∈ N : n has at most r prime divisors } ; LIYANG YANG R d := A d − π ( X ) ξd ; S := X n ∈A∩ P f w p := cw p , if p = P n or p ≥ x b/a ;min (cid:18) cw p , c − b − a log P n log x (cid:19) , otherwise, where ≤ b ≤ c ≤ a = cu and w p := 1 − u log p log x . W ( A , u, λ ) := X ♭s ∈A (cid:16) s,P (cid:16) X a (cid:17)(cid:17) =1 − λ X X a p X ca p | s f w p + X p > X a X h ∈A p | h S ( A ( β ) ∗ , z ) := X β X n ∈A ( β ) ∗ ( n,P ( z ))=1 where < a β , both are undetermined parameters.Define J ( λ ) := W ( A , u, λ ) − λ S ( A ( β ) ∗ , X − η ) . For simplicity, we shall denote by z := X a , y := X ca . Lemma 3. Assume that b = 1 or b > such that a ≥ c + b + 1 , then we have (2) S ≥ J ( λ ) if λ − < c − a. Proof. Notice that S = X ♭s ∈A∩ P (cid:16) s,P (cid:16) X α (cid:17)(cid:17) =1 O (cid:0) X − α (cid:1) , thus we only need the following inequality:(3) X ♭s ∈A\ P (cid:16) s,P (cid:16) X α (cid:17)(cid:17) =1 λ X ♭s ∈A (cid:16) s,P (cid:16) X α (cid:17)(cid:17) =1 X X α p X u f w p + λ X β X n ∈A ( β ) ∗ (cid:16) n,P (cid:16) X − η (cid:17)(cid:17) =1 O (cid:0) X − α (cid:1) , with the assumption that < ρ < α . To this end, we divide it into two cases: Case 1: s ∈ A \ P , so that s has at least prime factors. If s has a primefactor p which is larger than P s and log p log X ≤ b + 1 a − log P s log X , then X p | s f w p ≥ c − a log P s log X + c − b − a log P s log X = 2 c − b − ≥ c − a. Otherwise, every prime divisor of s which is larger than P s must satisfy log p log X ≥ b + 1 a − log P s log X , N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 5 which means that f w p ≥ c − a log p log X , for all p | s. This provides that X p | n f w p ≥ cω ( s ) − a log s log X ≥ c − a. thus we have (3) because of λ − < c − a. Case 2: s ∈ A ∩ P Similarly as above, we have P p | n f w p ≥ c − a. So (3)comes from the assumption that λ − < c − a. (cid:3) Therefore, we have Corollary 4. For λ − < c − a , if J ( λ ) := W ( A , u, λ ) − λ S ( A ( β ) ∗ , X − η ) ≫ π ( X ) ξ log X , then theorem 1 holds with τ = ρ. In the following sections, we will prove that J ( λ ) ≫ π ( X ) ξ log X and we can take ρ = . 3. Some auxiliary lemmas Lemma 5. For any x ≥ , we have Y p ≤ x (cid:18) − p (cid:19) = e − γ log x (cid:18) O (cid:18) x (cid:19)(cid:19) ; X p ≤ x p = log log x + c + O (cid:18) x (cid:19) , where c is an absolute constant.Remark. These two estimates are usually called Mertens formulas. Lemma 6 ([11]) . Let δ < and χ ( t ) be the characteristic function of interval ( − δ , δ ) extended to be periodic with period 1,then there exists A ( t ) , B ( t ) such that A ( t ) ≤ χ ( t ) ≤ B ( t ) where A ( t ) , B ( t ) can be written as A ( t ) := 2 δ − ( N + 1) − + X ≤| n |≤ N A n e ( nt ) ,B ( t ) := 2 δ + ( N + 1) − + X ≤| n |≤ N B n e ( nt ) , with coefficients A n , B n satisfying max {| A n | , | B n |} ≪ δ , for ≤ | n | ≤ N . Lemma 7. Suppose that ≤ α < β ≤ and ∆ > with < β − α , then thereexists a smooth function χ with the period 1 satisfying that: (1): χ ( x ) = 1 if α + ∆ ≤ { x } ≤ β − ∆ , χ ( x ) = 0 if { x } ≤ α or { x } ≥ β , and χ ( x ) ∈ [0 , otherwise. LIYANG YANG (2): χ ( x ) = β − α + P ≤| h |≤ ∆ − − ε c h e ( hx ) + O (∆) , where c h ≪ ε min { | h | , β − α − ∆ } . Moreover, the function g ( x ) := P ≤| h |≤ ∆ − − ε c h e ( hx ) is real.Proof. Fix ε > small enough. Then by ([17] Lemma 12, Chapter 1) we have χ ( x ) = β − α − ∆ + X | h |≥ ( a j cos 2 πjx + b j sin 2 πjx ) . Take c j = a j − ib j and c − j = a j + ib j for any j ∈ N ≥ , then by estimations from ([17])on a j and b j , we have c h ≪ ε min { | h | , β − α − ∆ , r | h | r +1 } for any arbitrary integer r . Take r large enough such that r ≤ ε and H := ∆ − r +1 r , then X | h |≥ H c h e ( hx ) ≪ X | h |≥ H r | h | r +1 ≪ r | H | r ≪ ∆ . Obviously, g ( x ) = P ≤| h |≤ ∆ − − ε ( a j cos 2 πjx + b j sin 2 πjx ) is a real function. (cid:3) Set S e w ( A ) := X s ∈A (cid:16) s,P (cid:16) X α (cid:17)(cid:17) =1 X X α p X u p | s f w p , then by a direct computation we have Lemma 8. S e w ( A ) = (1 − bc ) X X a ≤ p Our main goal in this section is to prove that(4) X d X α ξd max N X dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ξπ ( X ) X − η with α as large as possible.However, the lemmas in [6] can only give the result without taking max betweenthe two sums. We should point out that with some slight modifications of the proofin [6] we will be able to prove (4).This is a generalization of [6], Lemma 3: Lemma 9. Suppose X, M > , δ > , M a set of T integer points ( l, m ) with M m < M , λ lm real numbers for ( l, m ) ∈ M , and { a n } a sequence of complexnumbers, then X ( l,m ) ∈M max N X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N a n e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ D δ log (2 T X ) (cid:18) XM + δ − (cid:19) X n X/M | a n | , N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 7 where D δ = max ( l,m ) ∈M { ( l ′ , m ′ ) ∈ M : k λ lm − λ l ′ m ′ k < δ } . Proof. Define δ ( β ) := ( , if β γ, , otherwise , which is a truncation function. Then we have δ ( β ) = Z A − A e iβt sin γtπt d t + O (cid:18) A | γ − β | (cid:19) as in the proof of Lemma 2 of [16]. Here we take A = 2 T X , γ lm = log (cid:0) N lm + (cid:1) ,for ( l, m ) ∈ M . where p N lm = max n ∈ X : max N X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N a n e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn n a n e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Then we have X ( l,m ) ∈M max N X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N a n e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X ( l,m ) ∈M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N lm a n e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ( l,m ) ∈M Z A − A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n a n n it e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) sin γ lm tπt (cid:12)(cid:12)(cid:12)(cid:12) d t ! + O X ( l,m ) ∈M X n | a n | A log N lm +1 / mn ! ≪ X ( l,m ) ∈M Z A − A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n a n n it e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · min { γ lm , | t | } d t · log A + O X ( l,m ) ∈M X n | a n | A log N lm +1 / N lm ! ≪ log A · Z A − A X ( l,m ) ∈M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n a n n it e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · min { log X, | t | } d t + O X ( l,m ) ∈M X n | a n | A log X +1 / X ! ≪ log A · X ( l,m ) ∈M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n a n n it e ( λ lm n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · Z A − A min { log X, | t | } d t + O X ( l,m ) ∈M X n | a n | A log X +1 / X ! ≪ D δ log (2 T X ) (cid:18) XM + δ − (cid:19) Z log X d t + Z A t d t ! X n | a n | ≪ D δ log (2 T X ) (cid:18) XM + δ − (cid:19) X n | a n | , LIYANG YANG where the last step comes from [6], Lemma 3. (cid:3) This is a generalization of [6], Lemma 5: Lemma 10. Suppose ε > , X > R , J, M > , < q X , log | a | ≪ log X , ( a, q ) = 1 , then X r ∼ R max N X X j ∼ J X m ∼ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N e (cid:18) ajmnrq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X ε (cid:18) JXq + RJM + qR (cid:19) . Proof. By lemma 3 of [15] we obtain X r ∼ R max N X X j ∼ J X m ∼ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X mn N e (cid:18) ajmnrq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ log X ( JM ) ε X r ∼ R (cid:18) JX · ( r, a ) rq + JM + qR (cid:19) . Hence, it follows from the same estimates in lemma 5 of [6]. (cid:3) This is a generalization of [6], Lemma 7: Lemma 11. Suppose that ε > , X > R , L, M > , < q X , ( a, q ) = 1 and a ≍ q , max n LMqR , qMX o < , a n , b m ≪ X ε . Then X r ∼ R max N X X l ∼ L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∼ M b m X mn N a n e (cid:18) lmnaqr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X ε R (cid:18) L + RM (cid:19) (cid:18) MX + 1 M RL + R (cid:19) . Proof. The proof is essentially the same as that of lemma 7 of [6], with lemma 3 of[6] replaced by lemma 9 above. (cid:3) This is a generalization of [6], Lemma 8: Lemma 12. Suppose that X, R, L > , a ≍ q , ( a, q ) = 1 , ε > and T X R < q Using Vaughan’s identity we split the inner sum above into ≪ log N sumsof the form X m ∼ M X mn N a n b m e (cid:18) nalmdq (cid:19) , with either(I) a n = 1 or log n , M < X , b m ≪ X ε , or(II) a n , b m ≪ X ε , X < M < X .Sums of type (I) can be handled by lemma 10 and sums of type (II) by lemma11 and the estimate above follows. (cid:3) Corollary 13. We have X d X α ξd max N X dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ξπ ( X ) X − η . N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 9 Sieve estimates Let f , F and F be the limit functions occurred in Beta-Sieve, which are givenby the following definition: f ( s ) := A s − log( s − for s f ( s ) := A s − (cid:18) log( s − 1) + Z s − duu Z u − log( v − v dv (cid:19) for s F ( s ) := A s − for s F ( s ) := A s − (cid:18) Z s − log( v − v dv (cid:19) for s F ( s ) := A s − for s β + 1 , where A = 2 e γ , β = 4 . · · · , A = 43 . · · · are defined in [3], Chapter 11.We can, with a patient calculation, show that for s ∈ [ β + 1 , β + 2) , we have F ( s ) = s − (cid:18) A log β s − C + 2 A log ( s − 1) + 4 A log( s − (cid:19) − A (1 + s log( s − s ( s − , where C is determined by F ( β + 1) = A ( β +1) . As shown in Lemma 15 below,the level of distribution of A can be taken as θ = − ρ − ε. Henceforth, we take a = ϑθ and optimize ϑ to get a better upper bound of ρ. Take z = X a and y = X ca from now on. Remark. The limit functions f and F are actually defined by systems of differentialequations piecewise respectively. f is increasing rapidly and very close to its limit1 when s ≥ . While F is decreasing with limit 1. We should point out that inour situation, it turns out that θ − c > β + 1 since we require that b ≥ , whichleads c to be relatively small. Thus the above expression of F is invalid. We willdiscuss this matter in the next section.Denote by A := A e γ ≈ . , which will be used in the following section.In this section, we will prove the following theorem, which improves [6], Lemma1: Theorem 14. Let notations be defined as before and assume that b = 1 or b > such that a ≥ c + b + 1 , then for any δ ∈ (cid:2) bϑ , cϑ (cid:3) we have, J ( λ ) ≥ ae − γ (1 + o (1)) λξπ ( X )log X H δ ( ϑ, b, c ) , where (5) H δ ( ϑ, b, c ) = 2 e γ ( A δ ( ϑ ) b + B δ ( ϑ ) c + D δ ( ϑ ) + F δ ( ϑ, c )) , with A δ ( ϑ ) = − e − γ f ( ϑ ) + 12 e γ Z δ ϑ F ( ϑ (1 − s )) dss + 1 ϑ log 1 − δδ ; B δ ( ϑ ) = e − γ f ( ϑ ) − e γ Z δ ϑ F ( ϑ (1 − s )) dss − a I ( ρ ); D δ ( ϑ ) = 12 e γ H ( ϑ, ϑθ, ϑθ ) − δ log 1 − δδ + 2 ϑθa I ( ρ ); F δ ( ϑ, c ) = − ae − γ Z cϑ δ (cid:16) cs − ϑ (cid:17) F ( aθ − ϑs ) d s ! . To this end, we need the following lemmas. Lemma 15. We have S ( A , z ) > ξπ ( X ) V ( z ) (cid:0) f (6) + o (1) (cid:1) , where V ( z ) = e − γ log z (cid:0) o (1) (cid:1) , and z := X a as mentioned before.Proof. Take M ≍ dX η ξ in Lemma 6 then we have A d = X p Xd | ⌈ ap + bq ⌋k ap + bq k < ξ X p X k ap + bdq k < ξ d X p X χ (cid:18) ap + bdq (cid:19) = π ( x ) ξd + E ( A d ) + O (cid:18) ξπ ( X ) X − η d (cid:19) , where X p X X ≤| l |≤ M a l e (cid:18) ( ap + b ) lqd (cid:19) ≤ E ( A d ) ≤ X p X X ≤| l |≤ M b l e (cid:18) ( ap + b ) lqd (cid:19) with | a l | + | b l | ≪ ξd , ∀ ≤ | l | ≤ M. Therefore, by partial summation we have E ( A d ) ≪ max N X X X ≤| l |≤ M ( | a l | + | b l | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anlqd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ max N X ξd dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anlqd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence the density function of sequence A is g ( d ) = d ; and thus, by Jurkat-Richert’stheorem, we obtain S ( A , z ) > ξπ ( X ) V ( z ) (cid:16) f (4) + O (cid:16) (log X ) − (cid:17)(cid:17) + O ξπ ( X ) X − η + X d X α ξd max N X dY X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Then this lemma comes from corollary 13 since f (6) > . (cid:3) N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 11 Lemma 16. If < a < δ ′ < ca ≤ θ , let w = X δ ′ , then X z p Corollary 13 shows that the level of distribution of A is X θ . Hence byJurkat-Richert’s theorem, we have S ( A p , z ) ξπ ( X ) V ( z ) p (cid:18) F ( s p ) + O (cid:18) (log X α p ) − (cid:19)(cid:19) + O X d X θ /p | R pd | , where s p = log X θ p log z , and R pd ≪ ξpd max N X pdY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) anlpdq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since X z p Define e A := (cid:26) n (cid:24) an + bq (cid:23) : n ∈ [ z, X ] , p | (cid:24) an + bq (cid:23) , (cid:13)(cid:13)(cid:13)(cid:13) an + bq (cid:13)(cid:13)(cid:13)(cid:13) < ξ (cid:27) , then we have the following auxiliary lemma. Lemma 17. For d | P ( z ) and p ≥ z , we have e A d = Xξp g ( d ) + E ( X ; p, d ) , where g ( d ) := Y p | d (cid:18) p − p (cid:19) ,E ( X ; p, d ) ≪ qξτ ( d ) + Xpdq X d = d d ( q, d )( a, pd ) . Proof. Define J := { j : | j + b | qξ } . e A d = X | j | qξ X n ∈ [ z,X ] an + b ≡ j (mod pq ) n ( an + b ) ≡ jn (mod dq ) X j ∈ J X n ∈ [ z,X ] an ≡ j (mod pq ) an ≡ jn (mod dq ) X j ∈ J X n ∈ [ z,X ] an − j ≡ pq ) n ( an − j ) ≡ dq ) X j ∈ J X d = d d X n ∈ [ z,X ] an − j ≡ pq ) n ( an − j ) ≡ dq )( n,d )= d X d = d d X j ∈ J X n ∈ [ z/d ,X/d ] ad n − j ≡ pq ) n ( ad n − j ) ≡ d q )( n,d )=1 X d = d d X j ∈ J X n ∈ [ z/d ,X/d ] ad n − j ≡ pd q )( n,d )=1 X d = d d X j ∈ J ( ad ,pd q ) | j (cid:18) ϕ ( d ) d · X − zpdq ( ad , pd q ) + O (1) (cid:19) = X d = d d (cid:18) qξ ( ad , pd q ) + O (1) (cid:19) (cid:18) ϕ ( d ) d · X − zpdq ( ad , pd q ) + O (1) (cid:19) = X d = d d ϕ ( d ) d · ( X − z ) ξpd + O X d = d d qξ ( ad , pd q ) + X d = d d X · ( ad , pd q ) pdq ! , and thus lemma follows by noting that ( a, pd q )( d , pd q ) ≤ ( q, d )( a, pd ) . (cid:3) N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 13 Hence e A has a density function g ( d ) with V ( z ) := Y p z (cid:0) − g ( p ) (cid:1) = Y p z (cid:18) − p + 1 p (cid:19) = e − γ log z (cid:0) o (1) (cid:1) by Mertens estimate . We will use Beta-Sieve theory to e A to obtain an upper bound with a largerexponent of level of distribution. To this end, we shall compute its dimension asfollows: X p ≤ v g ( p ) log p = 2 X p ≤ v (cid:0) log pp − log p p (cid:1) = 2 log v + O (1) , for any v ≥ . Therefore, the sieve dimension is 2. Denote by θ the exponent of level of distri-bution of e A . Lemma 18. Assuming w p y and p is a prime number, where w = X δ ′ , a δ ca ≤ θ , then we have S ( A p , z ) Xξp V ( z ) (cid:16) F ( s ′ p ) + O (cid:16) (log X ) − (cid:17)(cid:17) + X d ≤ Xθ p E ( X ; p, d ) , where s ′ p := log (cid:0) X θ /p (cid:1) log z . Proof. We have S ( A p , z ) = X n ∈A p ( n,P ( z ))=1 1= { p ′ : z p ′ X, p | (cid:24) ap ′ + bq (cid:23) , (cid:13)(cid:13)(cid:13)(cid:13) ap ′ + bq (cid:13)(cid:13)(cid:13)(cid:13) < ξ, (cid:18) p ′ (cid:24) ap ′ + bq (cid:23) , P ( z ) (cid:19) = 1 } + { p ′ : p ′ < z, p | (cid:24) ap ′ + bq (cid:23) , (cid:13)(cid:13)(cid:13)(cid:13) ap ′ + bq (cid:13)(cid:13)(cid:13)(cid:13) < ξ, (cid:18) p ′ (cid:24) ap ′ + bq (cid:23) , P ( z ) (cid:19) = 1 } { n : z n X, p | (cid:24) an + bq (cid:23) , (cid:13)(cid:13)(cid:13)(cid:13) an + bq (cid:13)(cid:13)(cid:13)(cid:13) < ξ, (cid:18) n (cid:24) an + bq (cid:23) , P ( z ) (cid:19) = 1 } + O (cid:0) ξπ ( z ) (cid:1) = S ( e A , z ) + O (cid:0) ξπ ( z ) (cid:1) , We now meet a sifting problem of dimension two. By Beta-Sieve theory we have S ( A p , z ) S ( e A , z ) + O (cid:0) ξπ ( z ) (cid:1) Xξp V ( z ) F ( s ′ p ) + O (cid:18) log X θ p (cid:19) − !! + X d X θ /p E ( X ; p, d )+ O (cid:0) ξπ ( z ) (cid:1) = Xξp V ( z ) (cid:16) F ( s ′ p ) + O (cid:16) (log X ) − (cid:17)(cid:17) + X d X θ /p E ( X ; p, d ) and the last inequality holds because ξπ ( z ) ≪ XξV ( z )(log X ) − . This completes the proof. (cid:3) Lemma 19. If a δ ′ ca ≤ θ , let w = X δ ′ , then X w p ≤ y w p S ( A p , z ) ξπ ( X ) V ( z ) ae − γ α Z ca δ ′ (cid:18) s − u (cid:19) F ( a ( θ − s )) d s + o (1) ! + O (cid:18) qξX θ + ε + X ε q (cid:19) . Proof. From lemma 18 we obtain X w p ≤ y w p S ( A p , z ) XξV ( z ) X w p ≤ y w p p F ( s ′ p ) + O (log X ) − X w p ≤ y p + E A ( X ; w, y )= XξV ( z ) X w p ≤ y w p p F ( s ′ p ) + o (1) + E A ( X ; w, y ) , where E A ( X ; w, y ) := X w p ≤ y w p X d ≤ Xθ p E ( X ; p, d ) . Use the same method in Lemma 16 to handle P w p ≤ y w p p F ( s ′ p ) and we obtainthat X w p ≤ y w p p F ( s ′ p ) = Z u δ (cid:18) s − u (cid:19) F (cid:18) θ − s ) α (cid:19) d s + o (1) . As for E A ( X ; w, y ) , noting that for any < θ < , and for any B X θ /p ,we have X d ∼ B ( k, d ) = X c | k X d ∼ Bc =( k,d ) c = X c | k c X d ∼ Bc − ( d,kc − ) =1 ≪ X c | k B = Bτ ( k ) , by Abel transformation, X d ∼ B ( k, d ) d ≪ τ ( k ) , which illustrates(6) X d X θ /p ( k, d ) d ≪ τ ( k ) log X. Hence we conclude that E A ( X ; w, y ) ≤ X w ≤ p ≤ y X d ≤ X θ /p | E ( X ; p, d ) |≪ X w ≤ p ≤ y X d X θ /p qξτ ( d ) + Xpq X d = d d ( a, pd )( q, d ) d d ! ≪ qξ X w ≤ p ≤ y X θ + ε p + Xq X w ≤ p ≤ y X d d X θ /p ( a, pd ) pd · ( q, d ) d . N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 15 Noticing that (6), the lemma follows immediately. (cid:3) Thus we conclude our results above in a more general form: Given z = X α , y = X β and w = X δ ′ , where α ≤ δ ′ ≤ β ≤ θ , then we have X z ≤ p 181 + e − B ≈ . , while a ≥ by Theorem 22. Hence ba < . < δ if ρ ≥ , since actually we cantake δ = 23 − ρ − A − r A − A ! ± − . Therefore, we can only use a 2-dimensional sieve to the last term in Lemma 8. Lemma 20. We have S (cid:16) A ∗ , X − η (cid:17) (cid:0) I ( ρ ) + o (1) (cid:1) ξπ ( X )log X + O X ε X r X ν | R ∗ r ( β ) | , where (7) X r X ν | R ∗ r ( β ) | ≪ ξπ ( X ) X − η , with ν = − β − ρ − η and I ( ρ ) is defined by (8) I ( ρ ) := Z α d u u (1 − u − ρ ) Z − u u d u u Z − u − u u d u u (1 − u − u − u ) . Proof. This follows from [5], Theorem 8.3 and [6]. (cid:3) Remark. We shall use (7) to give some restrictions in Theorem 22. Proof of theorem 14. We have X p > X a X h ∈A p ≪ X p > X a π ( x ) ξp ≪ π ( x ) ξX a ≪ X − η ξ = o ( ξπ ( X ) V ( z )) . It comes from lemma 15, lemma 16 and lemma 19 that λ − W ( A , u, λ ) ≥ λ − S ( A , z ) − S e w ( A ) + o ( ξπ ( X ) V ( z ))= W ( A , u, λ ) − W ( A , u, λ ) + o ( ξπ ( X ) V ( z )) , where W ( A , u, λ ) := λ − S ( A , z ) − ( c − b ) X X a ≤ p It is obvious that by Corollary 4 and Theorem 14 we have: Theorem 21. The restriction from the main terms is given by ≤ b ≤ c ≤ a = ϑθ b = 1 or a ≥ c + b + 1 , if b ≥ bϑ ≤ δ ≤ cϑ max bϑ ≤ δ ≤ cϑ H δ ( ϑ, b, c ) > . where H δ ( ϑ, b, c ) is defined by (5) with F defined as before Theorem 14. Theorem 22. The restrictions from the error terms are given as the followinginequation systems: ( < ρ < min (cid:8) , a (cid:9) ,θ + ρ < , θ > . Proof. In Corollary 13 and Lemma 12 above, where we show that X r X α r max N X rY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N Λ( n ) e (cid:18) αnlrq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ π ( X ) X − η , with Y ≍ X ρ + η , we have to make sure that all the parameters satisfy the assump-tions of those lemmas. N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 19 Divide the intervals into dyadic segments and thus we have the following esti-mation: X i X i j X ε X T i + X (cid:18) T i R i (cid:19) ! ≪ X ε X + θ + ρ + η + X X i X i j X ρ + η ≪ X ε X + θ + ρ + η + X + ε + ρ + η , where L i j R i Y ≪ R i X ρ + η , T i ≪ R i X ρ + η and for simplicity we omit the preciserange of i and j , actually, only the bound i, j ≪ log X matters.Therefore, we get our restrictions as below: ( ε + + θ + ρ + η < − η, + ε + ρ + η < − η, i.e. ( θ + ρ < ,ρ < . Now let’s consider another estimation from (7). By assumption, we have X ρ + β + η We thus see that in our situation Laborde’s weight is not better thanRichert’s weight because of the effect from S ( A ( β ) ∗ , X − η ) , since I ( ρ ) a = a I ( ρ, a ) grows faster than f ( ϑ ) when ϑ ≥ . When b > , which forces that a ≥ c + b + 1 > , the contribution of S ( A ( β ) ∗ , X − η ) is just too large for our purpose. If we justtake δ = α as Harman did in [6], then by optimizing the parameters directly wehave τ < and we can take τ = . Estimates for exponential sums II Lemma 23 ([1]) . For any ι ∈ [0 , , let f h,ι,ς ( x ) := h ( x + ι ) γ + ςx, where h ∈ N and ς is an arbitrary constant. Take σ satisfying the restriction σ < γ − . Then any sufficiently small η > , we have min (cid:26) , X − γ H (cid:27) X h ∼ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∼ X Λ( n ) e ( f h,ι,ς ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ η X − σ − η , where H ≤ X − γ + σ + ε .Remark. We should point out that the O -constant is independent of ι and ς , namely,it’s uniform for ς , because only the behavior of f ′′ h,ς ( x ) is used when handling sumsof booth Type I and Type II, after using Heath-Brown’s identity (see [7]). This isa critical property as we will see in our situation we actually need to bound a meanestimate of the form X d ∼ D X l ∼ L | b l | X h ∼ H h max N ≤ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∼ N Λ( n ) e (cid:18) h ( n + ι ) γ + alnqd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We are showing the level of distribution is θ = γ − − ρ .In this section we aim to prove the following lemma: Lemma 24. For c ∈ (cid:0) , (cid:1) , θ = γ − − ρ, we have (9) X d ≤ X θ ξd max N ≤ X dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Nn ∈P Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ξπ c ( X γ ) X − η , where γ = c and P := {⌊ n c ⌋ : n ∈ N } .Proof. It is clearly that p = ⌊ n c ⌋ if and only if there exists a nonnegative ν < such that n c = p + ν , which, by a direct check, is equivalent to ⌊− p γ ⌋ − ⌊− ( p + 1) γ ⌋ = 1 , where γ is taken to be the inverse of c traditionally.Hence we can take φ ( n ) := ⌊− n γ ⌋ − ⌊− ( n + 1) γ ⌋ to be a characteristic functionof P , and thus for any N ≤ X , we have dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Nn ∈P Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N φ ( n )Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E ( N, d ) + E ( N, d ) , where(10) E ( N, d ) := dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N (( n + 1) γ − n γ ) Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and E ( N, d ) := dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N ( {− n γ } − {− ( n + 1) γ } ) Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We will see later that E ( N, d ) and E ( N, d ) are different types of exponential sums,and the former is algebraic, while the latter is analytic. Hence we use differentmethods to handle them respectively. N A DIOPHANTINE INEQUALITY INVOLVING A PRIME AND AN ALMOST-PRIME 21 Estimate of E ( N, d ) : Write E ( N, d ) in an integral form and integral byparts we have E ( N, d ) := dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N (( t + 1) γ − t γ ) d X n ≤ t Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z N (( t + 1) γ − t γ ) d dY X l =1 c l X n ≤ t Λ( n ) e (cid:18) anldq (cid:19) ≤ Z N max T ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dY X l =1 c l X n ≤ t Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) ( t + 1) γ − − t γ − + O ( 1 N ) (cid:19) dt ≪ max T ≤ N dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ T Λ( n ) e (cid:18) anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where c l = e iθ l , here θ l is the principle argument of the inner sum in (10).Thus by Lemma 12 we have Y X + θ ≪ π c ( X γ ) X − η , deducing that θ ≤ γ − − ρ − ε. Estimate of E ( N, d ) : Take η = 3 ε . By Lemma 7 we have E ( N, d ) = dY X l =1 c l X n ≤ N X ≤| h |≤ X − γ + σ + ε e ( h ( n + 1) γ − e ( hn γ ))2 πih Λ( n ) e (cid:18) anldq (cid:19) + O X γ − − σ dY X l =1 X n ≤ N Λ( n ) = dY X l =1 c l X ≤| h |≤ X − γ + σ + ε X n ≤ N Λ( n ) E ( N, d ) − E ( N, d )2 πih + O (cid:0) dY X γ − σ + η (cid:1) . where E ι ( N, d ) := dY X l =1 c l X ≤| h |≤ X ε h X n ≤ N Λ( n ) e (cid:18) h ( n + ι ) γ + anldq (cid:19) , for ι ∈ { , } . We split the summation range into dyadic segments, a typicalone is E ι j ( N, d ) := dY X l =1 c l X h ∼ H h X n ≤ N Λ( n ) e (cid:18) h ( n + ι ) γ + anldq (cid:19) ≪ dY X l =1 H X h ∼ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N Λ( n ) e (cid:18) h ( n + ι ) γ + anldq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where H is of the form j X − γ , and j ≪ log X since H ≤ X − γ + σ + ε .Hence by Lemma 23 we have X d ≤ X θ ξd max N ≤ X |E ( N, d ) | ≪ X ι ∈{ , } X j ≪ log X X d ≤ X θ ξd max N ≤ X (cid:12)(cid:12)(cid:12) E ι j ( N, d ) (cid:12)(cid:12)(cid:12) ≪ X d ≤ X θ ξY X γ − σ − η ≪ ξ π c ( X γ ) X θ − σ − η . So it suffices to take θ ≤ γ − − ρ .Combining the above discussion we thus obtain (9). (cid:3) Proof of Theorem 2 Denote by ˆ B := (cid:26)(cid:24) b + paq (cid:23) : p X, p ∈ P , (cid:13)(cid:13)(cid:13)(cid:13) b + paq (cid:13)(cid:13)(cid:13)(cid:13) < ξ (cid:27) , where P := {⌊ n c ⌋ : n ∈ N } . By taking M ≍ dX η ξ in Lemma 6 we have for and d ∈ N B d = X p X,p ∈P d | ⌈ ap + bq ⌋k ap + bq k < ξ X p X,p ∈P k ap + bdq k < ξ d X p X,p ∈P χ (cid:18) ap + bdq (cid:19) = π c ( X γ ) ξd + E ( B d ) + O (cid:18) ξπ c ( X γ ) X − η d (cid:19) , where X p X,p ∈P X ≤| l |≤ M a l e (cid:18) ( ap + b ) lqd (cid:19) ≤ E ( B d ) ≤ X p X,p ∈P X ≤| l |≤ M b l e (cid:18) ( ap + b ) lqd (cid:19) with | a l | + | b l | ≪ ξd , ∀ ≤ | l | ≤ M. As shown in Lemma 15, by partial summation we have E ( B d ) ≪ max N X X X ≤| l |≤ M ( | a l | + | b l | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N,p ∈P Λ( n ) e (cid:18) anlqd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ max N X ξd dY X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n N,p ∈P Λ( n ) e (cid:18) anlqd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , so the density function of sequence B is g ( d ) = d , and the corresponding level ofdistribution θ can be taken to be γ − − ρ .Since the level here is quite small, there might be little room for other sievetechniques. Thus we choose to use Laborde’s results to deal with B directly. Lemma 25. There are infinitely many P r in B if cθ ≤ r − . . Proof. This is essentially Theorem 3 of [10]. However, the upper bound for Λ therecan actually be taken to be 0.144, since log 6 − B − D B − log(1 + e − B )6 B ≈ . . So we can take . rather than . in the statement of Laborde’s theorem. 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Vinogradov, The method of trigonomrtrical sums in the theory of numbers , translatedrevised and annotated by K. F. Roth and Anne Davenport, Interscience Piblishers, Londonand New York, 1954. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China E-mail address ::