The existence of small prime gaps in subsets of the integers
aa r X i v : . [ m a t h . N T ] M a y THE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THEINTEGERS
JACQUES BENATAR
Abstract.
We consider the problem of finding small prime gaps in various sets
C ⊂ N . Followingthe work of Goldston-Pintz-Yıldırım, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting q n denote the n -th prime in C , we will establish thatfor any small constant ǫ >
0, the set { q n | q n +1 − q n ≤ ǫ log n } constitutes a positive proportion ofall prime numbers. Using the techniques developed by Maynard and Tao we will also demonstratethat C has bounded prime gaps. Specific examples, such as the case where C is an arithmeticprogression have already been studied and so the purpose of this paper is to present results forgeneral classes of sets. Introduction and framework
One of the most famous unsolved problems in Number Theory is the so-called Twin Prime Con-jecture, which posits the existence of infinitely many pairs of primes ( p, p ′ ) for which p − p ′ = 2.Throughout the past century a great amount of work has been done with regards to this conjectureand we refer the reader to [6] and [17] for some historical background on the subject. In this articlewe will build upon the methods developed by Goldston, Pintz, Yıldırım and more recently Maynardand Tao. Letting p n denote the n -th prime number, it was shown in [13] thatlim inf n →∞ ( p n + m − p n ) ≪ e m + ε for any ε > | { p n ≤ N | p n +1 − p n ≤ η log N } | ≥ C N log N for some constant C depending on η and any natural number N . In this paper we will investigatewhich conditions ensure that a set C ⊂ N will also have many small prime gaps. Definition 1.1.
Let q n denote the n -th prime number in C and for η > , write π C ( N ; η ) := | { q n ≤ N | q n +1 − q n ≤ η log N } | . We say that C frequently contains η -small prime gaps if there exists a constant C , depending on η ,such that π C ( N ; η ) ≥ C N log N for all positive integers N ≥ . We also define the quantity H C ( m ) := lim inf n →∞ ( q n + m − q n ) . From a probabilistic point of view, let us first show that a random subset of the primes, with positiverelative density ρ , frequently contains η -small prime gaps (for any η ). To avoid issues of indepen-dence, we will work with the sets P C ( N ; η ) ( i ) := { p n ≤ N | p n , p n +1 ∈ C , n ≡ i mod 2 and p n +1 − p n ≤ η log N } for i = 1 ,
2. For fixed N , denote π ∗C ( N ; η ) for the cardinality of the largest of these two sets. Lemma 1.2.
Let B be a random subset of the primes defined by letting the independent events p ∈ B occur with probability P ( p ∈ B ) = ρ . Writing λ ( N, η ) := ρ π ∗ N ( N ; η ) ≥ ρ π N ( N ; η ) / one hasthat P \ η [ k ≥ \ N ≥ k (cid:26) π ∗ B ( N ; η ) ≥ λ ( N, η )2 (cid:27) = 1 , (1) where η is made to run over the fractions (1 /n ) n ∈ N .Proof. For fixed N and η , write M := π ∗ N ( N ; η ) and let X ∼ B ( M, λ ( N, η )) denote a Binomialrandom variable ( i.e. a sum of M independent Bernoulli trials, each of which yields success withprobability ρ ). Observe that for any natural number k one has that P ( π ∗ B ( N ; η ) ≤ k ) ≤ k X j =0 (cid:18) Mj (cid:19) ( ρ ) j (1 − ρ ) M − j = P ( X ≤ k ) . It now follows after a simple application of Chernoff’s inequality (see for instance [20, Theorem 1.8]) that P ( X ≤ λ/ ≤ P ( | X − E ( X ) | ≥ λ/ ≤ e − λ and to conclude the proof of (1), we need only invoke the Borel-Cantelli lemma. (cid:3) Despite these heuristics one can piece together large subsets of the primes which have only largeprime gaps. From [8, Theorem 3] we get the existence of a constant
C > | { p n ≤ N | p n +1 − p n ≤ h } | ≤ C min (cid:18) h log N , (cid:19) π ( N )for any pair of positive integers N and h . In particular, any hope of obtaining a result of the form“If C ∩ P has positive relative density in the primes then C has small prime gaps” is dashed. Indeed,taking h = log N/ C in the statement just above, we gather that the collection of primes p n forwhich p n +1 − p n ≥ / C log p n constitutes a positive proportion of all prime numbers. It thusbecomes apparent that some kind of structure must be imposed on C if we wish to get primes inshort intervals. We will explore two scenarios in which we are able to control the interaction of C with arithmetic progressions. In each setting we provide a somewhat general result and then givesome examples of sets obeying the desired properties. The two main examples are Theorem 1.3 (Bohr sets have small prime gaps) . Let g ( x ) = P Dj =0 α j x j ∈ R [ x ] and suppose allcoefficients α j are Diophantine. Let d ∈ (0 , , η > and write { x } for the fractional part of a realnumber x . Then the Bohr set A := { n ∈ N | { g ( n ) } ∈ [0 , d ] } frequently contains η -small prime gaps and H A ( m ) ≪ d e m + ε for ε > arbitrary. HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 3
Theorem 1.4 (Shifted sets of square-free integers have small prime gaps) . Let η > and a ∈ N be arbitrary. Then the set of shifted square-free integers B := { n ∈ N | n + a is square-free } frequently contains η -small prime gaps. Remark 1.5.
In Theorem 1.3 one expects that the statement remains true as long as the coefficient α D is irrational, however with our current methods this seems out of reach. We will discuss this ingreater detail in section 6. In Theorem 1.4 we do not obtain a bound for H B ( m ) . This is becauseour method relies on the ability to establish the correct order of magnitude π B ( N ; η ) ≫ η N/ (log N ) and we are currently unable to do so for bounded prime gaps. The case where C is an arithmetic progression has been studied by several authors. These results,which will be stated in section 6, differ from our own in the sense that we obtain small prime gapsfrequently, as opposed to “infinitely often”. For other results concerning small gaps in special setsof primes we refer the reader to [1],[16], [19]. Notation
We introduce some standard notation that will be used throughout the paper. Forfunctions f and g we will use the symbols f ≪ g and f = O ( g ) interchangeably to express Landau’sbig O symbol. A subscript of the form ≪ η means the implied constant may depend on the quantity η . The statement f ∼ g means f and g are asymptotically equivalent, i.e., lim x →∞ f ( x ) /g ( x ) = 1.For two positive integers l, m we write l ≍ m when l/ ≤ m ≤ l and the superscript ♭ indicatesa summation over squarefree variables. We reserve the letter µ for the M¨obius function and Λ forthe von Mangoldt function. In place of Λ we will sometimes use the function θ ( n ) = (cid:26) log n if n is prime0 otherwise . The main results
Type A sets.
We first consider sets
A ⊂ Z which exhibit an even distribution amongarithmetic progressions of any given modulus. Fix a natural number k . We say that a k -tuple H = ( h , ..., h k ) has height h if each member is bounded in size by h . A set A will be of type A k ifit exhibits the following properties.(a) (Estimates for progressions in A )There exist constants c ( k ) , c ′ > , θ > N , X n ≤ N,n + H⊂A n ≡ a mod q c ( k ) Nq + R A ( N, a, q )and for any ε >
0, the remainder term satisfies the bound X q ≤ Q max a |R A ( N, a, q ) | ≪ ε N − c ′ (2) in the range Q ≤ N θ and uniformly over all k -tuples H of height h ≤ log N . JACQUES BENATAR (b) (Estimates for primes in A )There exists a constant c ( k ) > X n ≤ N,n + H⊂A Λ( n ) = c ( k ) N + O (cid:18) N (log N ) C (cid:19) (3) for any C >
0. In addition, for every pair
A, B >
0, we have the estimate X q ≤ Q ′ X χ mod qχ = χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Nn + H⊂A Λ( n ) χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ N (log N ) B (4) uniformly in the range Q ′ ≤ (log N ) A . The index χ mod q runs over Dirichlet characters ofmodulus q and χ denotes the trivial character.(c) (A bilinear form estimate for A )There exists a constant C = C ( A ) such that X q ≤ Q qϕ ( q ) ∗ X χ mod q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ M X l ≤ Lml + H⊂A a m b l χ ( ml ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ( M + Q ) / ( L + Q ) / (log M L ) C k a k k b k (5) for any pair of integers M, L and Q ≤ max( M θ , L θ ). The sum P ∗ χ indicates we are summingover primitive characters and k . k denotes the ℓ norm of a sequence.In this setting we will prove the following result. Theorem 2.1.
Any set
A ⊂ N of type A k frequently contains η -small prime gaps, for η > arbitrary. Assuming that c := sup l ∈ N c ( l ) /c ( l ) < ∞ , one has that H A ( m ) ≪ exp (2 m/θ + 2 c /θ + ε ) for all natural numbers m and ε > arbitrary. With some additional structure on A , we can obtain an improved result for type A sets. Definition 2.2.
Let G be a finite set of primes and write P ( G ) = Q p ∈G p . We say that a set Y ⊂ N is locally well distributed if it satisfies estimates of the type X n ≤ Nn ≡ a mod q Y ( n ) = λ Nq + R Y ( N, a, q ) for some λ > and there exists a constant ǫ > such that X q ≤ Q ( q, P ( G ))=1 max a |R Y ( N, a, q ) | ≪ N (log N ) A for Q ≤ N ǫ and A > arbitrary. Definition 2.3.
Let A be a subset of the natural numbers for which we can write A ′ + Y ⊂ A ,where A ′ is a type A set and Y is locally well distributed. Then we say A = ( A ; A ′ , Y ) is of typeA’. HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 5
Theorem 2.1’.
Let A be a type A ′ set. Then for any η > , A frequently contains η -small primegaps. In addition one has the estimate H A ( m ) ≪ exp (2 m/θ + 2 c (1) / ( c (1) θ ) + ε ) for all naturalnumbers m ε > arbitrary. The constants c (1) , c (1) are associated to A ′ . We will apply these theorems to the case where C is a Bohr set. In section 6 we will prove that,after a somewhat careful selection of k -tuples H , conditions ( a ) − ( c ) hold. From this, Theorem 1.3will follow. For linear Bohr sets (i.e. when D = 1) we will make use of Theorem 2.1’. It should alsobe noted that, trivially, N is of type A k for any k ∈ N , so that we recover the work of Goldston,Pintz and Yıldırım (e.g. [8, Theorem 1]).2.2. Type B sets.
We will call
B ⊂ Z a type B set if it has the form B = \ m ∈ M [ a ∈ M m { n ∈ N | n ≡ a mod m } , where M is a collection of pairwise co-prime integers and associated to each m ∈ M is a selectionof residue classes M m ⊂ Z m . We will write N m for the complement of M m and assume that X m ∈ M m ≥ x N m ϕ ( m ) ≪ x κ . for some κ > x ≥
1. In this particular setting we will show that
Theorem 2.4.
Let
B ⊂ N be a type B set satisfying the conditions listed above and let η > bearbitrary. Then B frequently contains η -small prime gaps. Remark 2.5.
Consider a type B set to which there are coupled infinitely many m ∈ M . To dealwith sums of the form P p B ( p ) , we are essentially counting primes in arithmetic progressions.However, since M contains infinitely many moduli and we are not able to process a large quantityof remainder terms, venturing down this avenue would pose a problem. Instead we shall work with“approximate type B sets B ( z ) ”, which are collections of the form B ( z ) := \ m ∈ M m To prove the main Theorems we will largely follow the framework set up in [6], [8] and [13]. Let H ⊂ N with |H| = k and for any such set, define the polynomial P H ( n ) := Y h ∈H ( n + h ) . Also, let l < k and write P ( z ) := Q p ≤ z p . From now on C will denote either a type A k set A oran approximate type B set B ( z ). In the latter case write F for the collection of primes dividing M ( z ) := Q m ∈ M ,m ≤ z m and for type A k sets we take F to be empty. When possible, we will treatboth types of sets in a unified manner.As in [8], the first part of Theorem 2.1 is proven by evaluating weighted sums of the form JACQUES BENATAR X H N X n =1( P H ( n ) , P ( R δ ))=1 X h ≤ hn + h ∈C θ ( n + h ) − log(3 N ) w ( n ) , (7)where N is a natural number, h := η log N and H runs over all k -tuples of height h . To obtain theestimates for H A ( m ) we may drop the condition ( P H ( n ) , P ( R δ )) = 1 and it becomes unnecessaryto some over all k -tuples. It should be noted that the expression (7) can only be positive if thereexists an interval of length h in which C has at least two primes. In addition, (7) can only be “large”if C has “many” small prime gaps. Next we observe that, when splitting the above expression intotwo parts, the condition n + h ∈ C falls only on the first sum. Intuitively, this indicates that anextra factor (the density of C ) will appear when dealing with this first sum but not for the secondsum. To overcome this issue we will consider carefully selected k -tuples H .The weight function w ( n ) was introduced in [13] as a generalization of the GPY sieveΛ R ( n, l ) := 1( k + l )! X d | P H ( n ) d ≤ R µ ( d ) (cid:18) log Rd (cid:19) k + l . Let H := { h , ..., h k } be an admissible k -tuple and suppose F : R k → R is a differentiable functionsupported on the simplex R k = n ( x , ..., x k ) ∈ [0 , k | P ki =1 x i ≤ o . Define λ d ,d ,...,d k := k Y i =1 µ ( d i ) d i ! ′ X r ,...,r k d i | r i µ ( Q ki =1 r i ) Q ki =1 ϕ ( r i ) F (cid:18) log r log R , ..., log r k log R (cid:19) , where the superscript ′ indicates the summation takes place over variables coprime to W . We willconsider the sieve weights w ( n ) := X d i | n + h i ( d , F )=1 λ d ,d ,...,d k with corresponding sums S := X n ∼ N ( P H ( n ) , P ( R δ ))=1 w ( n ) , S := k X i =1 X n ∼ N,n + h i ∈A ( P H ( n ) , P ( R δ ))=1 θ ( n + h i ) w ( n ) . To avoid the effect of small primes we let W := Q p ≤ D p for some D ( k ) and sum over values n ≡ a mod W for some ( a , W ) = 1. We will also assume λ d ,...,d k is supported on variablescoprime to W . After establishing a Bombieri-Vinogradov type result in section 3, we will prove twocrucial asymptotic estimates in section 4, namely Propositions 2.8 and 2.9, which are the analoguesof [8, Propositions 1 and 2]. In section 5 these will be used to obtain an asymptotic formula for(7). On the other hand we will demonstrate a connection between the quantity (7) and π C ( N ; η ),thereby proving our main results. HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 7 Definition 2.6. Let C be a type A k set A or an approximate type B set B ( z ) . We introduce theconstant S k ( C ) := Y p ∈F D Proposition 2.8. Fix k ∈ N and let C be a type A k set or an approximate type B set B ( z ) .Suppose N c ≪ R δ (log N ) − c for some constants c i depending on k . Furthermore, assume δ > is sufficiently small compared to k − , H ∈ h k is admissible and h ≪ log R with h → ∞ as N → ∞ .Then we have ♯ X n ≤ N,n + H⊂C ( P H ( n ) , P ( R δ ))=1 w ( n ) = γ ( C , H ) S k ( C ) N (log R ) k ϕ ( W ) k W k +1 I k ( F )(1 + O ( δk )) . When C = B ( z ) , the superscript ♯ indicates we are selecting those values of n satisfying n + H ⊂ Z × m ∩ M m for all m ∈ M , m ≤ z and n ≡ a mod W . The constant factor is given by γ ( C , H ) = Y m ∈ M D Let ε > be fixed. Given the same conditions as in Proposition 2, we have for N c ≪ R / − ε δ (log N ) − c and H := H ∪ { h } ∈ h k +1 admissible ♯ X n ≤ N,n + H⊂C ( P H ( n ) , P ( R δ ))=1 θ ( n + h m ) w ( n ) = γ ( C , H ) S k ( C ) N (log R ) k +1 ϕ ( W ) k W k +1 J ( m ) k ( F ) (cid:0) O ( δk + G ( H )) (cid:1) , when h m ∈ H and in the case h / ∈ H , one has ♯ X n ≤ N,n + H⊂C ( P H ( n ) , P ( R δ ))=1 θ ( n + h ) w ( n ) = γ ( C , H ) S k − ( C ) N (log R ) k ϕ ( W ) k − W k I k ( F ) (cid:0) O ( δk + G ( H )) (cid:1) . JACQUES BENATAR The remainder G ( H ) satisfies X H ∈ h k +1 h i distinct G ( H ) = o ( h k +1 ) as D → ∞ . A Bombieri-Vinogradov theorem for type A k sets In this section we make the necessary changes to the proof of the Bombieri-Vinogradov theorem toensure an identity of the form X ≤ m ≤ N,m + H⊂A m ≡ a mod q θ ( m ) = c Nϕ ( q ) + E ( N ; q, a ) , with the error term obeying the bound X q ≤ N θ − ε max a ( a,q )=1 | E ( N ; q, a ) | ≪ A,ε N (log N ) A (8)for any ε > A > 0. In the next section we will need this result to demonstrate Propositions2.8 and 2.9 for type A sets. Remark 3.1. To avoid any additional assumptions on type A k sets we wish to forgo the use of theP´olya -Vinogradov theorem. This result is used in the proof of the Bombieri-Vinogradov theoremand if one were to follow the proof word-for-word in our current setting, one would require a boundof the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N,n + H⊂A χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ε N / , for any non-trivial character χ mod q with < q ≤ N / − ε . For this reason, we will rearrange theargument in [2, Chapter 28 ] to better suit our purposes. Let us begin by defining the sums ψ A ( N, χ ) := X ≤ m ≤ Nm + H⊂A Λ( m ) χ ( m ) and ψ A ( N ; q, a ) := X ≤ m ≤ Nm + H⊂A m ≡ a mod q Λ( m ) = 1 ϕ ( q ) X χ mod q X ≤ m ≤ Nm + H⊂A χ ( a )Λ( m ) χ ( m ) . Removing the expected main term from ψ A ( N ; q, a ), we may write ψ A ( N ; q, a ) − c k Nϕ ( q ) = 1 ϕ ( q ) X χ mod q χ ( a ) ψ ′A ( N, χ )where ψ ′A ( N, χ ) = (cid:26) ψ A ( N, χ ) when χ = χ ψ A ( N, χ ) − c k N when χ = χ . For small moduli, e.g. q ≤ (log N ) A , we can estimate the sum X q ≤ Q ϕ ( q ) X χ mod q | ψ ′A ( N, χ ) | HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 9 immediately, using the conditions imposed in ( b ). The resulting contribution is bounded by theRHS of (8). An estimate for large moduli For large values of q ≤ Q we will use the estimate Proposition 3.2. For any ε > and A > , there exists a constant C > such that X q ≤ Q qϕ ( q ) max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ mod q χ ( a ) ψ ′A ( N, χ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:16) Q N / + N / Q + N (cid:17) (log QN ) C + QN (log N ) A (9) in the range Q ≤ N / − ε . Proof. We will rework the argument laid out in [2, chapter 28] in conjuction with the inequality(5).Once (9) is established, the proof of (8) is straightforward.With this in mind, we decompose our sum, as described in [2, p. 139], to obtain for χ = χ ψ ′A ( N, χ ) := X m ≤ Nm + H⊂A Λ( m ) χ ( m ) = X m ≤ N Λ( m ) f ( m ) = S + S + S + S where S = X n ≤ U Λ( n ) f ( n ) ≪ U, S = − X t ≤ UV X md = tm ≤ U,d ≤ V µ ( d )Λ( m ) X r ≤ N/t f ( rt ) ,S = X d ≤ V µ ( d ) X h ≤ N/d f ( dh ) log h and S = − X U ≤ m ≤ N/V Λ( m ) X V 1. To bound the sum S we firstrestrict the variable t to the range [1 , U ] and then to the range [ U, U V ]. Denote the resulting sumsas S = S ′ + S ′′ . The sums S ′ and S can be combined to create some degree of cancellation by noting that X χ mod q χ ( a ) ( S ′ + S ) = X t ≤ U − ( µ ⋆ Λ)( t ) X r ≤ N/t X χ mod q χ ( a ) f ( rt ) + µ ( t ) X r ≤ N/t X χ mod q χ ( a ) f ( rt ) log r = ϕ ( q ) X t ≤ U µ ( t ) log t X r ≤ N/trt ≡ a mod q f ( rt ) + µ ( t ) X r ≤ N/trt ≡ a mod q f ( rt ) log r = ϕ ( q ) X t ≤ U µ ( t ) X r ≤ N/trt ≡ a mod q f ( rt ) log( rt )= ϕ ( q ) X t ≤ U ( t,q )=1 µ ( t ) c k Ntq log N − c k Ntq − Z N R ( x, tq ) x dx ! . where ⋆ denotes a Dirichlet convolution and the last identity follows from summation by parts. Forthe sum P χ mod qχ = χ ( S ′ + S ) we simply subtract the χ = χ term in the first line. The additionalterm will be of the same form as the last line, just above, and can be dealt with in the same way.This last expression may be recast as c ϕ ( q ) q X l | q µ ( l ) X t ≤ Ut ≡ l µ ( t ) t ( N log N − N ) + O Z N ϕ ( q ) X t ≤ U ( t,q )=1 R ( x, tq ) dxx . (10)To estimate (10) we require the following lemma. Lemma 3.3. Let Φ c ( x ) := exp((log x ) c ) , then for any positive, squarefree integer l ≤ Φ / ( x ) , onehas the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ≤ xt ≡ l µ ( t ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ω ( l ) ω ( l ) Φ / ( x ) , where ω ( l ) denotes the number of primes dividing l .Proof. The bound is easily demonstrated by induction on ω ( l ). To be precise, we will show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ≤ xt ≡ l µ ( t ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (2 ω ( l )) ω ( l ) exp − (cid:18) log (cid:18) x (Φ / ( x )) ω ( l ) (cid:19)(cid:19) / ! . (11)When ω ( l ) = 0 and hence l = 1, this estimate is obtained as a consequence of the bound | P n ≤ x µ ( n ) | ≪ x/ Φ / ( x ) (see for instance [21, Chapter II.5]) followed by partial summation.Let us now suppose that (11) has been established whenever ω ( l ′ ) < r and let l be such that ω ( l ) = r . Furthermore, given j ≤ r , let l j denote a generic divisor of l having j prime factors. Then HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 11 we have X t ≤ xt ≡ l µ ( t ) t = µ ( l ) l X t ≤ x/l ( t,l )=1 µ ( t ) t = µ ( l ) l X j ≤ r − ( − j (cid:18) rj (cid:19) X t ≤ x/lt ≡ l j µ ( t ) t + X t ≤ x/lt ≡ l µ ( t ) t = µ ( l ) l X t ≤ x/lt ≡ l µ ( t ) t + O r − l X j ≤ r − (cid:18) rj (cid:19) ( r − j exp " − (cid:18) log (cid:18) xl (Φ / ( x )) r − (cid:19)(cid:19) / = µ ( l ) l X t ≤ x/lt ≡ l µ ( t ) t + O r − l r r exp " − (cid:18) log (cid:18) xl (Φ / ( x )) r − (cid:19)(cid:19) / . Repeating this process K times yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ≤ xt ≡ l µ ( t ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ l K + 2 r − ω ( l ) ω ( l ) X m ≤ K l m exp " − (cid:18) log (cid:18) xl m (Φ / ( x )) r − (cid:19)(cid:19) / , which gives the desired estimate after selecting the smallest value K for which l K ≥ Φ / ( x ). (cid:3) In order to estimate the first sum in (10) we apply the previous lemma for values of l ≤ Φ / ( N )(combined with the fact that ω ( l ) ≪ log l ) and use a trivial bound otherwise. Together with (2) itfollows that X q ≤ Q ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ mod q ( S ′ + S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ N (log N ) A for any A > 0. To estimate S and S ′′ we will make use of the bilinear form inequality (5). Since,currently, we are considering sums which run over complete sets of characters for each modulus q ,these must first be converted to sums involving only primitive characters.With this in mind, let χ mod q be generated by the primitive character χ mod q ′ and write a m = Λ( m ) , b k = X d | kd ≤ V µ ( d ) . First observe that X U ≤ m,k ≤ N/Umk ≤ N,mk + H⊂A a m b k [( χ − χ )( mk )] = X U ≤ m,k ≤ N/Umk ≤ N a m b k f ( mk ) − X U ≤ m,k ≤ N/Umk ≤ N, ( mk,q )=1 a m b k f ( mk ) . With regards to S , it follows that the contribution made by the character χ to the LHS of (9)does not exceed ♭ X q ≤ Qq ≡ q ′ ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N, ( mk,q )=1 a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ♭ X q ≤ Qq ≡ q ′ ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X l | q µ ( l ) X U ≤ m,k ≤ N/Umk ≤ N,l | mk a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ♭ X q ≤ Qq ≡ q ′ ϕ ( q ) X l | qq ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N,l | mk a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ♭ X l ≤ Q ♭ X q ≤ Qq ≡ q ′ l ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N,l | mk a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and hence we get that ♭ X q ≤ Qq ≡ q ′ ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N, ( mk,q )=1 a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ♭ X l ≤ Q log Nϕ ( l ) 1 ϕ ( q ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N,l | mk a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Summing over all primitive characters we conclude that S accounts for a total bound of T := ♭ X l ≤ Q log Nϕ ( l ) ♭ X q ′ ≤ Q ϕ ( q ′ ) ∗ X χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N,l | mk a m b k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ♭ X l ≤ Q log Nϕ ( l ) ♭ X q ′ ≤ Q ϕ ( q ′ ) ∗ X χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N a m b k f ( mk ) X ( d ,d ) ∈ P ( l ) d ,l ( m )1 d ,l ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ♭ X l ≤ Q log Nϕ ( l ) X ( d ,d ) ∈ P ( l ) ♭ X q ′ ≤ Q ϕ ( q ′ ) ∗ X χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X U ≤ m,k ≤ N/Umk ≤ N a m d ,l ( m ) b k d ,l ( k ) f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In the last two lines we have used 1 d,l ( m ) to indicate those integers m for which ( l, m ) = d and P ( l ) := (cid:8) ( d , d ) ⊂ N | d | l, d | l and l | d d (cid:9) . The three inner-most sums can now be dealt withas in [2], except that the estimate (5) takes on the role of the large sieve. Indeed, by decomposingthe range of k into dyadic intervals [ M, M ] and setting c m := a m d ,l ( m ), h k := b k d ,l ( k ) we find HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 13 that on each such interval X q ≤ Q qϕ ( q ) ∗ X χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X M ≤ m ≤ Mm ∈ [ U,N/U ] U ≤ k ≤ N/m c m h k f ( mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ( M + Q ) / ( NM + Q ) / ( X m ≤ M Λ( m ) ) / ( X k ≤ N/M d ( k ) ) / log( N ) C ≪ ( M + Q ) / ( NM + Q ) / ( M log M ) / ( NM (log N ) ) / log( N ) C ≪ ( Q N / + QNM / + QN / M / + N )(log N ) C +2 . The sum S ′′ may be treated as S to obtain X q ≤ Q qϕ ( q ) ∗ X χ ( | S ′′ | + | S | ) ≪ (cid:18) Q N / + QNU / + QN U / V / + N (cid:19) (log N ) C . (12)Combining all of the above, we get X q ≤ Q qϕ ( q ) ∗ X χ max y ≤ N | ψ A ( x, χ ) | ≪ (cid:18) Q N / + N + QNU / + U QN / + Q / U (cid:19) (log QU N ) C + QN (log N ) A since U = V . In the range N / ≤ Q ≤ N / we choose U = N / /Q . In this case it follows easilythat all terms involving U are bounded by Q N / . When Q ≤ N / we select U = N / . For such values of Q we obtain the bound QN / . Thisconcludes the proof of (9). To finish the proof of (8) we estimate the sum X (log N ) A ≤ q ≤ Q ϕ ( q ) max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ mod q χ ( a ) ψ ′A ( N, χ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) by decomposing the range of q into dyadic intervals and applying (9) to each of the resulting sums.4. Obtaining the asymptotics Let q = R β denote a prime with 0 ≤ β ≤ δ . In order to prove Proposition 2.8, it is enough todemonstrate that A := ♯ X n ≤ Nn + H⊂C w ( n ) ∼ γ ( C ) S k ( C ) N (log R ) k ϕ ( W ) k W k +1 I k ( F )and for each 1 ≤ l ≤ k , A ( q, l ) := ♯ X n ∼ N,n + H⊂C q | n + h l w ( n ) ≪ S k ( C ) N (log R ) k ϕ ( W ) k W k +1 I k ( F ) (cid:18) βϕ ( q ) + 1 ϕ ( q ) (cid:19) . To complete the argument one then sums the latter bound over all primes D < q ≤ R δ .These estimates are obtained by combining the ideas in [8, Proposition 1] and [13, Sections 4 and 5] with some added details which we will point out. Given an approximate type B set B ( z ), let M ( z ) = Q q ∈ M q For q = 1 or q prime, 1 ≤ l ≤ k and H = { h , h , ..., h k } , we introduce the quantities y r ,...,r k ( q, l ) = k Y i =1 µ ( r i ) ϕ ( r i ) ! X d ,...,d k r i | d i , ( d i , F )=1 q | d l λ d ,...,d k Q ki =1 d i , (13) x r ,...,r k ( q, l ) = k Y i =1 µ ( r i ) g ( r i ) ! X d ,...,d k r i | d i , ( d i , F )=1 q | d l λ d ,...,d k Q ki =1 ϕ ( d i )(14)and the analogous sums y ′ r ,...,r k ( q, l ), x ′ r ,...,r k ( q, l ) in which the condition q | d l is replaced by q ∤ d l .Here, g and ρ are the totally multiplicative functions given by g ( p ) = p − ρ ( p ) = (cid:26) p | Q ki =1 ( h − h i )1 otherwise . on primes p . We also set y max := sup r ,...,r k | y r ,...,r k | . Proposition 4.1. As N → ∞ one has the asymptotic formulas A ( q, l ) = NW X r i Q ki =1 ϕ ( r i ) y r ,...,r k ( q, l ) + y r ,...,r k ( q, l ) y ′ r ,...,r k ( q, l ) + ( y ′ r ,...,r k ) ( q, l ) q ! + B ( q, l )(15) A ( i )2 ( q, l ) = Nϕ ( W ) X r i Q ki =1 g ( r i ) x r ,...,r k ( q, l ) + x r ,...,r k ( q, l ) x ′ r ,...,r k ( q, l ) + ( x ′ r ,...,r k ) ( q, l ) ϕ ( q ) ! + B ( q, l )(16) and the errors are bounded by B ( q, l ) ≪ y max N (log R ) k ϕ ( q ) W D , B ( q, l ) ≪ N y max (log N ) A + ( y ( i ) max ) N ( ϕ ( W ) k − )(log R ) k g ( q ) W k − D . HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 15 Proof. Expanding A ( q, l ) we find that A ( q, l ) = X d ,...,d k e ,...,e k λ d ,...,d k λ e ,...,e k ♯ X n ∼ N,n ∈A q | n + h l [ d i ,e i ] | n + h i γ ( C ) NW ∗ X d ,...,d k e ,...,e k q | [ d l ,e l ] λ d ,...,d k λ e ,...,e k Q ki =1 [ d i , e i ] + ∗ X d ,...,d k e ,...,e k q ∤ [ d l ,e l ] λ d ,...,d k λ e ,...,e k q Q ki =1 [ d i , e i ] + R , where the superscript ∗ indicates that ( d i , F ) = 1 and ( e i , F ) = 1 for all i and we have used theimportant fact that n + h i ∈ M ∩ Z × m for all i to ensure that the constant γ ( C ) appears. In theevent that q | M ( z ), clearly A ( q, l ) vanishes and hence we may assume q ∤ F . The error satisfies R ≪ X d ,...,d k e ,...,e k | λ d ,...,d k λ e ,...,e k | ≪ λ max ( X d 0. For the second inequality we used the trivial estimate E ′ ( N, t ) ≤ N (log N ) /ϕ ( t ).As with the discussion of A ( q, l ) we split the two sums in the main term into four parts, dependingon the divisibility of d l , e l by q . The remainder of the proof goes through as in [13, Lemma 5.2]. (cid:3) To complete the proof of Propositions 2.8 and 2.9 we need the following sieve theory estimates. Lemma 4.2. Let γ be a multiplicative function satisfying ≤ γ ( p ) p ≤ − A − L ≤ X w ≤ p ≤ z γ ( p ) log pp − κ log( z/w ) ≤ A for all ≤ w ≤ z . Let g be the totally multiplicative function defined by g ( p ) = γ ( p ) p − γ ( p ) on primesand let G : [0 , → R be piecewise differentiable with G max := sup x ∈ [0 , ( | G ( x ) | + | G ′ ( x ) | ) . Thenfor any prime q = R β , we have the estimates ♭ X d This is a rearrangement of [9, Lemma 4]. The LHS of (18) produces a main term withintegral ranging in [log q/ log z, 1] rather than the full interval [0 , G max log q log z on the RHS of (18). (cid:3) Choosing the smooth weights y u ,...,u k := F (cid:16) log u log R , ..., log u k log R (cid:17) , we first observe that y u ,...,u k ( q, l ) = (cid:26) y u ,...,u k if q | u l − y u ,...qul,...,uk ϕ ( q ) if q ∤ u l and y ′ u ,...,u k ( q, l ) = (cid:26) q | u l y u ,...,u k − y u ,...qul,...,uk ϕ ( q ) if q ∤ u l . It follows that A ( q, l ) = NW X u ,...,u k Q ki =1 ϕ ( u i ) y u ,...,u k ( q, l ) + 2 y u ,...,u k ( q, l ) y ′ u ,...,u k ( q, l ) + ( y ′ u ,...,u k ) ( q, l ) q ! + E and the sum in the main term becomes X u i q | u l y u ,...,u k Q ki =1 ϕ ( u i ) + X u i q ∤ u l Q ki =1 ϕ ( u i ) y u ,...,u k q − /q ) ϕ ( q ) y u ,...,u k y u ,...qu l ,...,u k + y u ,...qu l ,...,u k /qϕ ( q ) ! Applying Lemma 4.2 to the summation over each variable u i we conclude the discussion of A ( q, l ).Before carrying out the estimate for A ( q, l ) we require a combinatorial rearrangement of the func-tions x ( q, l ) and x ′ ( q, l ). Lemma 4.3. For a given k + 1 - tuple H = { h , h , ..., h k } with h = h m one has that x r ,...,r k ( q, l ) = X a m y r ,...,a m ,...r k ( q, l ) ϕ ( a m ) + O (cid:18) y max ϕ ( W )(log R ) W D (cid:19) and the corresponding formula for x ′ r ,...,r k ( q, l ) is obtained by replacing y r ,...,a m ,...r k ( q, l ) with y ′ r ,...,a m ,...r k ( q, l ) . Furthermore, when h = h m for all ≤ m ≤ k , one has that x r ,...,r k ( q, l ) = h k +1 y r ,...,r m ,...r k ( q, l ) + O (cid:18) y max ϕ ( W )(log R ) W D B ( H ) (cid:19) , with the error B ( H ) satisfying X h ,h ,...,h k distinct B ( H ) ≪ (cid:26) h k +1 /ϕ ( q ) if q | r l h k +1 if q ∤ r l . Proof. We will focus on the second identity since the first is derived in much the same manner. Forthe same reason we will only discuss the case where q | r l . Let us begin by inserting the identity λ d ,...,d k = k Y i =1 µ ( d i ) d i ! X r ,...,r k d i | r i y r ,...,r k Q ki =1 ϕ ( r i )into the definition of x r ,...,r k ( q, l ) to get x r ,...,r k ( q, l ) = k Y i =1 µ ( r i ) g ( r i ) ! X d ,...,d k r i | d i k Y i =1 ρ ( d i ) µ ( d i ) d i ϕ ( d i ) ! X a ,...,a k d i | a i y a ,...,a k Q ki =1 ϕ ( a i ) . Setting s i := ( a i , F ) and interchanging the order of summation we first note that X d i r i | d i ,d i | a i ρ ( d i ) µ ( d i ) d i ϕ ( d i ) = µ ( r i ) ρ ( r i ) r i ϕ ( r ) Y p | a i p ∤ r i ,s i (cid:18) − p ρ ( p ) p − (cid:19) = µ ( r i ) ρ ( r i ) r i h ( a i ) ϕ ( r i ) h ( r i ) h ( s i ) . Here, h is the multiplicative function for which h ( p ) = 1 − ρ ( p ) p/ ( p − 1) on primes. We gather that x r ,...,r k ( q, l ) = k Y i =1 r i g ( r i ) ρ ( r i ) ϕ ( r i ) h ( r i ) ! X a ,...,a k r i | a i y a ,...,a k k Y i =1 h ( a i /s i ) ϕ ( a i ) . Taking into consideration the support of y we see that the only non vanishing terms occur when a i = r i or a i > r i D . Let us examine the latter situation. Suppose a j > r j D , and for each1 ≤ l ≤ k write t l for the largest divisor of a l /s l satisfying ρ ( t l ) = 0. Then we have X h ,h ,...,h k distinct X a ,...,a k r i | a i h ( a i ) ϕ ( a i ) ≪ k X h ,h ,...,h k distinct X a ,...,...,a k t ...t k | h − h a j >r j D ϕ ( a l /t l s l ) ϕ ( t l ) . Switching the order of summation in the last expression and recalling that q divides r l (and hence a l )we get the desired error term. The main term becomes Q ki =1 ( g ( r i ) r i ρ ( r i ) /ϕ ( r i ) ) y r ,...,r m ,...r k ( q, l ).After noting that g ( p ) p/ϕ ( p ) = 1 + O ( p − ) and summing over all k + 1- tuples H the resultfollows. (cid:3) The derivation of the estimate for A ( q, l ) is similar to our discussion of A ( q, l ). When h = h m ,equation (16) combined with the previous lemma leads to the sum X r i i = m Q ki =1 g ( r i ) X a m y r ,...,a m ,...r k ( q, l ) ϕ ( a m ) ! . We note that a factor (cid:18) ϕ ( W ) W Q p ∈F p>D ϕ ( p ) p (cid:19) is introduced after applying Lemma 4.2 to the expres-sion in brackets. Together with the factor appearing in equation (16), this accounts for the constant γ ( C , H ) in Proposition 2.4. The summations over the remaining variables are carried out as in [13,Lemma 6.3]. HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 19 Completing the proofs of the main theorems To complete the proofs of our main theorems we use the following key result. Proposition 5.1. Define the quantity M k := sup F P km =1 J ( m ) k ( F ) I k ( F ) , where the supremum is taken over all differentiable functions F : R k → R supported on the simplex R k = n ( x , ..., x k ) ∈ [0 , k | P ki =1 x i ≤ o . Then M k > log k − k − for sufficiently large k .Proof. This is [13, Proposition 4.3]. (cid:3) To prove the first part of Theorems 2.1, 2.1’ and Theorem 2.4, we recall that π C (2 N ; η ) = | { q n ≤ N | q n +1 − q n ≤ η log N } | and consider the sum˜ S R, C := 1 h (log R ) k X N ≤ n ≤ Nn ∈A (Θ C ( n, h ) − log(3 N )) (cid:3) X H w ( n ) , where Θ C ( n, h ) = X h ≤ hn + h ∈C θ ( n + h )and the superscript (cid:3) indicates that the summation takes place over admissible k -tuples H ⊂ [1 , h ]for which ( P H ( n ) , P ( R δ )) = 1 and H ∈ h k +1 . In the case of a type A ′ set, we will also assume thateach member of H lives in Y . The upper bound for ˜ S R, A is obtained in precisely the same manneras [8] with the addendum that X N ≤ n ≤ N Θ( n,h ) ≥ / N A ′ ( n ) ≤ h π A (2 N ; η ) + O ( N exp( − c p log N )) . Observe that when ( P H ( n ) , P ( R δ )) = 1, w ( n ) ≪ k log Rδ log N (log R ) k . It follows, as in [8], that˜ S R, C ≪ ( h π C (2 N ; η ) + o ( π (2 N ))) / × (log R ) k log NN / h k (2 k + 2)!2 (2 k log 3 N ) /δ log R (cid:18) hδ log R (cid:19) k/ (cid:18) hδ log R (cid:19) ( k +2) / . On the other hand, combining the asymptotics with Proposition 5, one finds˜ S R, C ≫ c S k ( C ) ϕ ( W ) k W k +1 log N I k ( F ) (cid:18) M k log R log N + c η Wϕ ( W ) D − c c + O ( k δ ) + O (cid:18) D ( k ) (cid:19)(cid:19) . In the case of a type A ′ set one gets a constant of the form c ( G , λ ) cη instead of cη . By choosing δ ≪ ηθ we get 1 + h/ ( δ log R ) ≪ h/ ( δ log R ). For η sufficiently sufficiently small and k sufficientlylarge, it is at once clear that ˜ S R > π A (2 N ; η ) ≫ C C ( k, z ) η r ( k ) π ( N ) , for some positive integer r ( k ). To finish the proof of Theorem 2.4 we now let z := C C ( k ) / ( κ +1) .Nowset D := B ( z ) c ∩ B and apply the Brun-Titchmarsh Theorem (see for example [11, Theorem 6.6])to obtain π D ( N ; η ) ≪ π D ( N ) ≪ X q ∈ M q ≥ z N q ϕ ( q ) π ( N ) ≪ π ( N ) z κ . For the second part of Theorem 2.1 we note that X N ≤ n ≤ Nn ∈A k X i =1 θ ( n + h i ) − log(3 N ) ! w ( n ) ≫ c S k ( A ) ϕ ( W ) k W k +1 log N I k ( F ) (cid:18) M k log R log N − c c (cid:19) . In order to make this sum positive we need only take k ≫ exp (2 m/θ + 2 c /θ + ε ).6. Some examples Arithmetic progressions. As a first example, we consider arithmetic progressions { n ∈ N | n ≡ a mod q } where ( a, q ) = 1. It is at once clear that the required conditions for a type B set hold and hencewe get the following corollary. Proposition 6.1. Let η > be arbitrary. Then any arithmetic progression ( nq + a ) n ∈ N with ( a, q ) = 1 frequently contains η -small prime gaps. Several authors have studied the case of arithmetic progressions and obtained results similar to theabove. Firstly, Goldston, Pintz and Yıldırım showed in [7] that there are small prime gaps in theprogression { n ≤ N | n ≡ a mod q } and one can even let q grow slowly with N . Theorem 6.2. Let ε and A be arbitrary fixed positive numbers. Let q and N be arbitrary, sufficientlylarge integers, satisfying q ( A, ǫ ) ≪ q ≪ (log log N ) A , N > N ( A, ǫ ) , and let a be arbitrary with ( a, q ) = 1 . Then there exist primes p, p ′ ∈ [ N/ , N ] such that p ′ ≡ p ≡ a mod q and p ′ − p < ε log p . T. Freiberg demonstrated, in [4], that it is possible to find consecutive primes in short intervalswhich are both congruent to a mod q . Theorem 6.3. Fix any positive number ǫ , and fix a pair of coprime integers q ≥ and a . Thereis an absolute positive constant c such that, for all sufficiently large X , X p r ≤ X,p r +1 − p r <ǫ log p r p r ≡ p r +1 ≡ a mod q ≥ X − c/ log log X . HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 21 Shifted sets of k -free numbers. Our second application of Theorem 2.4 pertains to shiftedsets of k -free numbers, i.e sets of the form B := \ p (cid:8) n ∈ N | n a mod p k (cid:9) , where a is any fixed integer. Again, it is easily verified that B is a type B set so we obtain Theorem1.4.6.3. Bohr sets. Sequences of the type ( { g ( n ) } ) n ∈ N , with g a polynomial, have been the subject ofmuch study. It was demonstrated by H.Weyl that they are uniformly equidistributed in the unitinterval, provided that the leading coefficient of g is irrational. Later on, I.M. Vinogradov showedthat the sequence remains equidistributed if one restricts n to prime values. In this section we willadd yet another result to this subject by proving Theorem 1.3. Let g ( x ) = P Dj =1 α j x j ∈ R [ x ]. ABohr set is a collection of the form A := { n ∈ N | { g ( n ) } ∈ [0 , d ] } . Definition 6.4. An irrational number α is of type ρ > if ρ = sup n γ ∈ R | lim inf m →∞ m γ k mα k = 0 o . Here k x k denotes the distance from x to the nearest integer. A number α which obeys such a boundis said to be Diophantine. Remark 6.5. Observe that Theorem 1.3 holds for Lebesgue-almost all D -tuples ( α , ...α D ) ∈ R D and as a consequence of the Thue-Siegel-Roth Theorem (see [3] ), whenever the α j are algebraicirrationals.We would like to show that A is a type A k set (for large k ) but there are some immediate algebraicobstructions that must be overcome. Consider, for example, the polynomial g ( x ) = √ x . Observethat the events n ∈ A , n + 1 ∈ A , n + 2 ∈ A and n + 3 ∈ A are not independent, since n √ n o − n √ n + 1) o + 3 n √ n + 2) o − n √ n + 3) o ≡ . In other words, (2) does not even hold for -tuples. Some background information and tools . We begin by recalling some useful facts fromthe theory of Diophantine approximation (which can be found in [22]). For the remainder of thissection we will assume α is Diophantine of type ρ . Given any positive integer M , the collection { m ≤ M | { mα } ∈ [0 , d ] } has a very neat combinatorial structure. In order to describe this struc-ture, rearrange the natural numbers up to M in such a way that { s α } < { s α } < ... < { s M α } .One has the recurrence relationship s j +1 = s j + s when s j ≤ M − s s j + s − s M when M − s < s j < s M s j − s M when s M ≤ s j . Without loss of generality we may assume || s α || ≤ || s M α || so that || s M α || , || ( s − s M ) α || ≥ / M .By Dirichlet’s theorem we have that || s α || ≤ /M and since α is Diophantine, we easily find that s ≫ ε M /ρ − ε for any small ε > 0. It also follows easily from the above that (cid:26) b ≤ M | k bα k ≤ M c (cid:27) ⊂ { b ≤ M | b ≡ s } (19) for any c > M sufficiently large). Another important notion related to our problem is thatof discrepancy. Given a sequence x = ( x , ..., x N ) in [0 , 1] and real numbers 0 ≤ δ < β ≤ 1, write A ([ δ, β )) = |{ n ≤ N | x n ∈ [ δ, β ) }| . The discrepancy of x is defined to be D N ( ω ) = sup ≤ δ<β ≤ (cid:12)(cid:12)(cid:12)(cid:12) A ([ δ, β ); N ) N − ( β − δ ) (cid:12)(cid:12)(cid:12)(cid:12) . Remark 6.6. Instead of working with the discontinuous characteristic function χ [0 ,d ] we shallsometimes make use of a smooth cut-off function ψ ( x ) : [0 , → [0 , satisfying ψ ( x ) = (cid:26) δ ≤ x ≤ d − δ x / ∈ [0 , d ] . Employing such a bump function will improve the rate of convergence of ψ ’s Fourier expansion.More precisely, if ψ is r -times continuously differentiable with ψ i (0) = ψ i ( d ) = 0 for i < k , oneeasily shows that the Fourier coefficients c k grow like c k ≪ min (cid:18) k , N rC k r (cid:19) provided that we choose δ := N − C . In other words, for such a choice of δ we get rapid convergenceas soon as k ≫ N C . The following results will aid us in the verification of conditions ( a ) − ( c ). For more details, we referthe reader to [12], [14], [10] and [18] respectively. Theorem 6.7 (Erd¨os-Tur´an-Koksma) . Let x = ( x , ..., x M ) be a sequence of real numbers in theunit interval [0 , . Then for any H ∈ N the discrepancy of this sequence is bounded by D M ( x ) ≪ H + X r ≥ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n M X n =1 e ( rx n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Theorem 6.8 (Weyl’s inequality) . Let f ( x ) ∈ R [ x ] be a polynomial of degree k with leading coeffi-cient α satisfying | α − r/ν | ≤ /ν for some pair of coprime integers r, ν with ν > . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N e ( f ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ N ε (cid:18) ν + 1 N + νN k (cid:19) − k for any ε > . Theorem 6.9. Let f ∈ R [ x ] be a polynomial of degree k , with leading coefficient α and suppose | α − a/v | ≤ /v with ( a, v ) = 1 and v ≤ N . Then for any ε > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 Λ( n ) e ( f ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ N ε (cid:18) N / + 1 ν + νN k (cid:19) − k . In addition we will use a special case of van der Corput’s lemma. HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 23 Lemma 6.10 (van der Corput, special case) . Let φ ( x ) ∈ C ([ a, b ]) such that | φ ′ ( x ) | ≥ and φ ′ ( x ) is monotonic in ( a, b ) . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba e ( φ ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (1) , with the implied constant being independent of a and b . Lemma 6.11. For α Diophantine and ε > fixed, one has the inequality |B α ( M, M k + C , m ) | = (cid:12)(cid:12)(cid:8) b ≤ M | k mb k α k ≤ /M k + C (cid:9)(cid:12)(cid:12) ≪ k,ε M − c ( C,ρ ) for any C > and uniformly in the range ≤ m ≤ M C − ε .Proof. Let ℘ km ⊂ N denote the set of all integers of the form mn k and write H := (cid:6) N C (cid:7) . Accordingto (19) there exists an integer s ≤ M k H such that (cid:12)(cid:12) B α ( M, M k + C , h ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:8) ls | l ≤ M k H/s (cid:9) ∩ ℘ kh (cid:12)(cid:12) . Since α is Diophantine, we have a lower bound of the type s ≥ ( M H ) c ( ρ ) . Combining this factwith the above estimate, the result easily follows. (cid:3) Verifying conditions ( a ) − ( c ) for Bohr sets . To begin with, we will need to choose acollection h of k - tuples H = ( h , ..., h k ) in such a way that the problem arising in Remark 6.6 canbe avoided. Consider the first k primes p , ..., p k and for each p i , let e i be the smallest naturalnumber for which 1 /p e i i < d . We will say H ∈ h if for each p i , one has that p e i i | h j ∀ j = i and( p i , h i ) = 1.In order to demonstrate a bound of the type stated in condition (2), we will consider sums of theform X n ≤ N,n ≡ a mod qn + h i ∈A i , with each A i ⊂ A . The sets A i will vary for each N but we will ensure that |A i | ≫ k N independentlyof N . Under these circumstances all of the arguments leading to the proofs of our main theoremsgo through.Finally, we observe that in the case of a linear Bohr set A ( N, d ) := { n ≤ N | { nα } ∈ [0 , d ] } wehave the bonus of an additive structure, since A ( N, d/ 2) + A ( N, d/ ⊂ A ( N, d ). Proposition 6.12. Subject to the constraints described above, one has an inequality of the form X q ≤ Q |R A ( N, a, q ) | ≪ N − c ( ρ ) for Q = N / and some constant c > depending on ρ .Proof. Let ε > g ( n )) n ∈ N is uniformly equidistributed mod 1. As a result, there exists an interval I ⊂ [ d/ , d ] oflength | I | ≤ d/ N ε containing no more than N − ε elements of the form g ( n ) with n ≤ N . We maynow replace d with the right end-point of I , since reducing d by a factor 2 will have no bearingon the statement of Theorem 13. Moreover, for each individual i ≤ k we shorten the interval I ,obtained above, at its left-end and modify the constant term in g , so that we may assume d = p − e i i . In other words, for each i ≤ k we will work on a shortened interval I i := [0 , d i ]. Letting A i denotethe Bohr set associated to I i , we note that k Y i =1 A ( n + h i ) = k Y i =1 [0 ,d i ] ( g ( n + h i )) = k Y i =1 ψ ( g ( n + h i )) + O k X r =1 [ d i − δ,d i ] ( g ( n + h i )) ! , where δ = N − ε and ψ is the truncated Fourier series, as described in Remark 6.6. To deal with theremainder term we invoke the Erd¨os-Tur´an-Koksma inequality (setting H = N ε ) and gather that X q ≤ Q |R A ( N, a, q ) | ≪ X q ≤ Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | m |≤ N ε m =0 c m X n ≤ N/q e k X i =1 g ( nq + a + h i ) m i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20) + X q ≤ Q N ε X r =1 r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N/q e ( g ( nq + a + h i ) r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( N − ε ) , where the subscript m runs over k -tuples ( m , ..., m k ), the Fourier coefficients obey the bound c m = Q m i =0 c m i ≪ Q m i =0 | /m i | and | m | denotes the maximum norm of m . Moreover, when p e i i | m i for all component m i of m , we may assume that P i m i = 0. To see this, observe that when p e i i | m i , the Fourier coefficient c m ≤ /N kε and the number of k -tuples m for which P i m i = 0 isat most N ( k − ε .Due to our choice of H , we now see that the polynomial P ( x ) := P ki =1 g ( xq + a + h i ) m i appearingin (20) does not collapse to a constant function and in fact has a leading coefficient of the form β ( q, m ) := (cid:0) Dj (cid:1) q D − j α k − j ( P ki =1 h ji m i ) for j = 0 or 1. We will deal with the first sum in (20) andnote that the second quantity is treated in very much the same way. As a consequence of Weyl’sinequality, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N/q e ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:18) Nq (cid:19) − c ( ε ) as long as one can find natural numbers N ε ≤ ν ≤ N D − ε and r ≥ ν , satisfying (cid:12)(cid:12)(cid:12) β ( q, m ) − rν (cid:12)(cid:12)(cid:12) ≤ ν . First consider those “good” pairs ( q, m ) for which β ( q, m ) admits a rational approximation asabove. Applying Weyl’s inequality to these pairs, it follows that their contribution to the RHS of(20) is easily subsumed in the desired bound. Also, since α is Diophantine, we can choose ε in sucha way that ( q, m ) is good whenever q D ≤ N ε .Let us arrange the remaining “bad” pairs ( q, m ) into dyadic cubes Q ≤ q ≤ Q and H ≤ | m | ≤ H .By Dirichlet’s theorem, for each such bad pair, we thus find a ν ≤ N c ( ε ) / such that | β − r/ν | ≤ /νN D − ε . In other words, we can write q ∈ [ ν ≤ N c ( ε ) / (cid:26) q ≤ Q| || q D | m | α || ≤ N D − ε (cid:27) . HE EXISTENCE OF SMALL PRIME GAPS IN SUBSETS OF THE INTEGERS 25 From the previous lemma we gather that each set in the union above has size O (cid:0) Q − c ( ρ ) (cid:1) (for ε sufficiently small) so that X Q≤ q ≤ Q X H ≤| m |≤ H ( q, m ) bad c m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N/q e ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (log N ) Q − c ( ρ ) N Q . The proof is completed by summing over all dyadic cubes for which Q ≥ N ε/D .When D = 1, the proof proceeds in the same manner except it is no longer necessary to sum over k -tuples m in equation (20). (cid:3) Lemma 6.13. For Q = (log N ) A , we have X q ≤ Q ϕ ( q ) ∗ X χ mod q | ψ ′A ( N, χ ) | ≪ N (log N ) A Proof. Let ε > ψ ′A ( N, χ ) using the smooth cut-off function discussed in theproof of Proposition 6.12. For χ = χ this yields ψ ′A ( N, χ ) = X | m |≤ N ε c m X n ≤ N χ ( n )Λ( n ) e k X i =1 g ( n + h i ) m i ! + O ( N − ε )= X | m |≤ N ε c m τ ( χ ) q X a =1 X n ≤ N χ ( a )Λ( n ) e k X i =1 g ( n + h i ) m i + anq ! + O ( N − ε )where τ ( χ ) = P qm =1 χ ( m ) e ( m/q ), c m ≪ / | m | and H ≪ N ε . First consider the case D = 1. Since α is Diophantine, we may estimate the exponential sum immediately, using Theorem 6.9 to obtainthe desired bound.When D ≥ 2, we adopt the exact same strategy as in the proof of Proposition 6.12. Organise m into dyadic cubes and then consider separately the good and bad pairs ( q, m ).Finally, when m = 0, we have the classical estimate X n ≤ N χ ( n )Λ( n ) ≪ N (log N ) A . The case χ = χ is treated similarly. (cid:3) Our final task is to prove the bilinear form estimate (5). We recall the following variant of theclassical large sieve inequality X q ≤ Q qϕ ( q ) ∗ X χ max u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ M X l ≤ Lmn ≤ u a m b l χ ( ml ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ( M + Q ) / ( L + Q ) / ( X m ≤ M | a m | ) / ( X l ≤ L | b l | ) / log(2 M L ) . (21) Lemma 6.14. X q ≤ Q qϕ ( q ) ∗ X χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ M X l ≤ Lml + H⊂A a m b l χ ( ml ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ( M + Q ) / ( L + Q ) / (log M L ) k a k k b k . (22) Proof. We will assume without loss of generality that L ≥ M . The inner-most double sum in (22)can be expanded to obtain S := X m ≤ M X l ≤ Lml + H⊂A a m b l χ ( ml ) = X m ≤ M X l ≤ L a m b l χ ( ml ) X | r |≤ L C c r e k X i =1 g ( ml + h i ) r i ! + O (1 /L )= X | r |≤ L C c r X m ≤ M X l ≤ L a m b l χ ( ml ) "Z F r ( ml )( ML ) e ( x ) dx + e (cid:0) ( M L ) (cid:1) + O (1 /L ) . (23)Here F r ( x ) = P ki =1 g ( x + h i ) r i and we shift the leading coefficient by a large integer multipleof 2 πi to ensure that F r ( y ) ≥ ( M L ) for y ≥ 1. Furthermore, the constant C depends onlyon η , r = ( r , ..., r K ) ∈ Z K and regarding the Fourier coefficients, we have c r ≪ Q r i =0 | /r i | .Interchanging the order of summation and integration in (23) leads to the quantity Z L C η,K ) ( ML ) e ( x ) X m ≤ M X l ≤ LF r ( ml ) >x a m b l χ ( ml )(24)The condition F r ( ml ) > x can be replaced by an expression of the form ml ∈ ∪ i [ Z i ( F r − x ) , Z i +1 ( F r − x )] where Z i ( G ) denotes the i -th real zero of the function G . It therefore suffices to evaluate (24)under the new condition Z i ( F r − x ) < ml or rather H ( x ) := log( Z i ( F r − x )) < log( ml ). A simpleapplication of the Implicit Function Theorem shows that H ′ ( x ) ≪ /x , with the implied constantbeing independent of the modifications made to F r . Substituting the discontinuous integral (p. 165of [2]) 2 Z T − T e ( αt ) sin(2 πβt ) dtt = (cid:26) O ( T − ( β − | α | ) − ) if | α | < βO ( T − ( β − | α | ) − ) if | α | > β into (23), we are left with sums of the form Z T − T sin(2 π [log( m ) + log( l )] t ) Z L C η,K ) ( ML ) e ( x + tH ( x )) X m ≤ M X l ≤ L a m b l χ ( ml ) dx dtt . Choosing T = ( M L ) / , we see that ddx ( x + tH ( x )) is bounded away from zero and piecewisemonotonic so that van der Corput’s lemma can be applied. Note that, after applying a simple trigidentity, we have effectively separated the variables m and l so that the resulting quantities may beabsorbed into a m and b l . We may now proceed to use (21). 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