aa r X i v : . [ m a t h . N T ] M a r THE PRIMES ARE NOT METRIC POISSONIAN
ALED WALKER
Abstract.
It has been known since Vinogradov that, for irrational α , the sequenceof fractional parts { αp } is equidistributed in R / Z as p ranges over primes. Thereis a natural second-order equidistribution property, a pair correlation of such frac-tional parts, which has recently received renewed interest, in particular regardingits relation to additive combinatorics. In this paper we show that the primes do notenjoy this stronger equidistribution property. Introduction
Let
A ⊂ N be an infinite sequence of natural numbers, and let A N denote the first N elements of A . For α ∈ [0 , α A , taken modulo 1. Recallthat the sequence α A is equidistributed in R / Z if for every interval I ⊂ R / Z one haslim N →∞ N X x ∈ A N I ( αx ) = | I | . (1)For many arithmetic sequences A of interest, the sequence α A is equidistributed in R / Z for all irrational α . This is true for A = N itself, or more generally the set of k th powersfor any k ∈ N , and, most pertinently for us, the set of primes.In this paper we will consider a strictly stronger notion of equidistribution. Withnotation as above, we define the pair correlation function F ( A , α, s, N ) := 1 N X x i ,x j ∈ A N x i = x j [ − s/N,s/N ] ( α ( x i − x j )) , (2)where both the interval [ − s/N, s/N ] and the sequence α A are considered modulo 1.Informally, F ( A , α, s, N ) counts the number of pairs ( αx i , αx j ) such that the dis-tance αx i − αx j mod 1 is approximately s times the average gap length of the sequence αA N mod 1. Analysing the behaviour of F ( A , α, s, N ) for a specific α can require del-icate Diophantine information about α (see [7], [10]), but one may instead settle forresults which hold for almost all α .In the setting of (1), any A satisfies the equidistribution property for almost all α (the sharpest results in this direction are due to Baker [3]). However, in the setting ofpair correlations, the situation is more subtle. We say that the sequence A is ‘metricPoissonian’ if for almost all α ∈ [0 , s >
0, we have F ( A , α, s, N ) = 2 s (1 + o A ,α,s (1)) (3)as N → ∞ . Notice that if we had picked N i.i.d. random variables ( X n ) n ∈ [ N ] uniformlydistributed on R / Z , instead of the sequence αA N mod 1, then as N tends to infinity theequivalent pair correlation function would tend to 2 s with high probability. Therefore(3) may be viewed as some strong indication that α A mod 1 exhibits the behaviour ofa random sequence. The connection to the equidistribution property (1) was recentlymade rigorous: indeed, three simultaneous papers [1, 4, 12] recently showed that if (3) holds, for some fixed α and for all s , then for the same α one has that α A is equidis-tributed in R / Z .One might expect that, for the classical sequences A where α A is equidistributedfor irrational α , one could prove that these sequences A are metric Poissonian. Indeed,for k > A the set of k th powers this was shown by Rudnick and Sarnak [9].However, the sequence A = N is not metric Poissonian. This follows from considerationof the continued fraction expansion of α , but is in fact a special case of a more generalphenomenon, connected to the large additive energy of this particular set A .For a finite set B ⊂ N we define the additive energy E ( B ) to be the number ofquadruples ( b , b , b , b ) ∈ B such that b + b = b + b . If | B | = N , then we havethe trivial bounds N ≪ E ( B ) N . For x ∈ R , let us write k x k for min y ∈ Z | x − y | .Then, for an additive quadruple ( b , b , b , b ) satisfying b + b = b + b , obviously if k α ( b − b ) k sN then k α ( b − b ) k sN . This is extremely different behaviour thanthat which would be seen if αb , αb , αb , αb were genuinely i.i.d. uniform randomvariables on R / Z , and indeed we have the following result of Bourgain, which showsthat all sets of nearly maximal energy fail to have the metric Poissonian property. Theorem 1. [2, Appendix]
Suppose limsup N →∞ E ( A N ) N > . Then A is not metric Poissonian. It is clear that the sequence A = N satisfies the hypotheses of this theorem, and thereforethis sequence is not metric Poissonian.Remarkably, a near-converse to this theorem has also been proved to be true. Theorem 2.
Let δ > be fixed, and suppose that E ( A N ) ≪ δ N − δ for this fixed δ andfor every N . Then A is metric Poissonian. This theorem first appears as stated in the recent work of Aistleitner, Larcher andLewko [2]. It immediately implies the theorem of Rudnick-Sarnak on k th powers, andalso earlier work on lacunary sequences [11].It is natural to wonder whether there is a tight energy threshold for this problem.Although the truth seems unlikely to be so clean, it is certainly interesting to considerthe behaviour of specific sets A which satisfy N − ε ≪ ε E ( A N ) ≪ o ( N ) for all ε > ). Theorem 3 (Main Theorem) . The primes are not metric Poissonian.
When A is the set of primes one has E ( A N ) ≍ N (log N ) − , so certainly the primesare not included in the range of applicability of either Theorem 1 or Theorem 2.In [2], Bourgain constructs a sequence A which is not metric Poissonian but nonethe-less has E ( A N ) = o ( N ), thereby showing that the converse to Theorem 1 is false.A quantitative analysis of his argument shows that E ( A N ) ≪ ε N (log log N ) − + ε isachievable, for any ε >
0. So, as an immediate corollary to Theorem 3, we have animproved bound for the smallest energy E ( A N ) of the initial segments of a set A whichis not metric Poissonian. The same result may be deduced from Theorem 3.2 of Harman’s earlier book [6], combined withthe relevant modification of the variance estimate from page 69 of the same volume. at the ELAZ 2016 conference in Strobl. HE PRIMES ARE NOT METRIC POISSONIAN 3
Acknowledgements
The author would like to thank Prof. R. Nair for making him aware of the centralquestion of this paper, Prof. C. Aistleitner for comments on an earlier version, andProf. B. J. Green for his doctoral supervision. A helpful conversation was also had withSam Chow. The work was completed while the author was a Program Associate at theMathematical Sciences Research Institute in Berkeley, which provided excellent workingconditions. The author is supported by EPSRC grant no. EP/M50659X/1.2.
Proof of Theorem 3
The plan of the proof is as follows. For each fixed α , we will try to find infinitely many n such that k αn k is extremely small. Using such an n we will be able to construct a scale N and a small constant s such that k αmn k s/N for some initial segment of integers m . By a variant of a well-known result concerning the exceptional set for the Goldbachproblem, we may show that many such mn are represented many times as p i − p j for twoprimes p i , p j p N , the N th prime. Combining all these observations will enable us toconclude, provided s is small enough, that F ( P , α, s, N ) > c for some constant c > s .Since this holds for infinitely many N , we cannot have F ( P , α, s, N ) = 2 s (1 + o α,s (1))for all almost all α and for all s >
0. In fact, we will show that, for almost all α , thisasymptotic fails to hold.We now begin to consider the details of this argument. We will use the followingresult of Harman on Diophantine approximation ([6, Theorem 4.2], [5]). Theorem 4.
Let ψ ( n ) be a non-increasing function with < ψ ( n ) . Suppose that X n ψ ( n ) = ∞ . Let B be an infinite set of integers, and let S ( B , α, N ) denote the number of n N , n ∈ B , such that k nα k < ψ ( n ) . Then for almost all α we have S ( B , α, N ) = 2Ψ( N, B ) + O ε (Ψ( N ) (log Ψ( N )) ε ) (4) for all ε > , with implied constant uniform in α , where Ψ( N ) = X n N ψ ( n ) and Ψ( N, B ) = X n Nn ∈B ψ ( n ) . This theorem may be thought of as a flexible version of Khintchines’s theorem on Dio-phantine approximation, in which one can further pass to approximations coming froma set B , provided B is relatively dense. The quality of the error term in this theoremis much better than we need in our application, although it is important that there isno dependence on N except through Ψ( N ). Earlier results of this type include a log N factor in the error, which would not have been adequate.The other technical tool will be the standard bound on the size of the exceptional setin a Goldbach-like problem. Theorem 5.
For a large quantity X , and natural number n X , define r ( n ) := X p i ,p j Xp i − p j = n log p i log p j . ALED WALKER
Then for any
B > , and for all but O B ( X log B X ) exceptional values of n X , we havethe approximation r ( n ) = S ( n ) J ( n ) + O B ( X log B X ) , (5) where S ( n ) := Q p > (cid:16) − p − (cid:17) Q p | np > p − p − n even, n oddis the singular series, and J ( n ) = ∞ Z −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Z e ( βu ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( − βn ) dβ is the singular integral.Proof. This result follows by trivial modifications of the usual argument for the binaryGoldbach problem, originally due (independently) to van der Corput, Estermann, and˘Cudakov. The clearest reference is [13, Chapter 3.2], or, for a more modern approach,one may consider the proof of Theorem 19.1 in [8, Chapter 19]. (cid:3)
We combine these two key ingredients in the following proposition.
Proposition 6.
There exists a small absolute c > , such that for almost all α ∈ [0 , ,and for all fixed s > , there exist infinitely many n satisfying:(i) k αn k < sn log n (ii) At least c log n of the numbers n , n , · · · , ⌊ log n ⌋ n are expressible in at least c n log n ways as the difference p − p of two primes p , p n log n .Proof of Proposition 6. Let c > c , let B be the set of natural numbers n which satisfy (ii), and let ψ ( n ) = min( , sn log n ). It isto this B and this ψ that we will apply Theorem 4.We claim that B is relatively dense. Indeed, let K be a large integer, and let n and m be natural numbers restricted to the ranges K n < K and 1 m ⌊ log 2 K ⌋ .For notational convenience we let X denote the quantity K log K , and we considerTheorem 5 with this X . We say that the pair ( n, m ) is exceptional if nm lies in theexceptional set from Theorem 5 for which the asymptotic formula (5) fails to hold. [Notethat nm X , so Theorem 5 applies in this setting.]The map ( n, m ) nm is at most log 2 K -to-1, due to the restricted range of m .Since the exceptional set from Theorem 5 has size at most O B ( K log B K ), for all B > O B ( K log B K ) exceptional pairs ( n, m ), for all B > − O B (log − B K )) K values of n ∈ [ K, K ) such that theasymptotic formula for r ( nm ) holds for all m log 2 K . Let D K denote this set of n . D K is certainly very dense in [ K, K ), and we claim further that D K ⊂ B , providedwe choose c small enough. Combining the different scales K will allow us to show that B is suitably dense. Indeed, let us analyse the asymptotic formula for r ( nm ). When nm is even, the singular series S ( nm ) is always Ω(1). By Fourier inversion, the singularintegral is exactly ( [0 ,X ] ∗ [ − X, )( nm ), which is Ω( X ) provided that | nm | X . But,by the choice of ranges for n and m , this inequality is always satisfied. So, for all n ∈ D K and for all m log n , such that nm is even, we have r ( nm ) ≫ X . HE PRIMES ARE NOT METRIC POISSONIAN 5
Removing the log weights on the primes, and recalling the definition of X , in partic-ular we notice that there is some small absolute constant c such the following holds: if n ∈ D K and if nm is even, there are at least c K/ log K pairs of primes ( p i , p j ) with p i , p j K log K such that p i − p j = nm . By the definition of B , provided c is chosensmaller than min( c , ), we have that D K ⊂ B .We may now prove that Ψ( B , N ) ≫ s Ψ( N ) for large N . Indeed,Ψ( B , N ) = X n Nn ∈B min( 12 , sn log n ) ≫ − O s (1) + ⌊ log N ⌋− X k = k X k n< k +1 n ∈B s k log(2 k ) ≫ − O s (1) + ⌊ log N ⌋− X k = k X k n< k +1 (1 − O B ( k − B )) sk k ≫ − O s (1) + ⌊ log N ⌋ X k =1 sk ≫ s log log N ≫ s Ψ( N ) . Therefore, applying Theorem 4 to this set B and this function ψ , the main termfrom the conclusion of Theorem 4 dominates the error term, and we conclude that foralmost all α there are infinitely many n ∈ B satisfying k αn k < sn log n . The propositionis proved. (cid:3) With this moderately technical proposition proved, the deduction of Theorem 3 isextremely short.
Proof of Theorem 3.
Let Ω ⊂ [0 ,
1] be the full-measure set of α for which Proposition 6holds. Let c be the constant from Proposition 6, and fix some s satisfying 0 < s < c .Let α ∈ Ω, and fix a large N to be one of the infinitely many natural numbers whichsatisfy the conclusions of Proposition 6.By construction, we know that k αN k < sN log N .
Therefore, for all d log N , we have k αdN k < sN . But by the second conclusion of Proposition 6, this implies that there are at least c N pairs of distinct primes p i , p j N log N such that k α ( p i − p j ) k < sN . Since P N ∼ N log N , and N is large, this certainly implies that F ( P , α, s, N ) > c > s. This holds for infinitely many N , and therefore for all α ∈ Ω we have F ( P , α, s, N ) = 2 s (1 + o (1))as N → ∞ . Since Ω has measure 1, Theorem 3 is proved. (cid:3) ALED WALKER
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