A pair correlation problem, and counting lattice points with the zeta function
aa r X i v : . [ m a t h . N T ] O c t A PAIR CORRELATION PROBLEM, AND COUNTING LATTICEPOINTS WITH THE ZETA FUNCTION
CHRISTOPH AISTLEITNER, DANIEL EL-BAZ, MARC MUNSCH
Abstract.
The pair correlation is a localized statistic for sequences in the unit interval.Pseudo-random behavior with respect to this statistic is called Poissonian behavior. Themetric theory of pair correlations of sequences of the form ( a n α ) n ≥ has been pioneeredby Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and ( a n ) n ≥ is an integersequence, often of arithmetic origin. Recently, a general framework was developed whichgives criteria for Poissonian pair correlation of such sequences for almost every real number α , in terms of the additive energy of the integer sequence ( a n ) n ≥ . In the present paper wedevelop a similar framework for the case when ( a n ) n ≥ is a sequence of reals rather thanintegers, thereby pursuing a line of research which was recently initiated by Rudnick andTechnau. As an application of our method, we prove that for every real number θ >
1, thesequence ( n θ α ) n ≥ has Poissonian pair correlation for almost all α ∈ R . Introduction and statement of results
A sequence ( y n ) n ≥ of real numbers is called uniformly distributed (or equidistributed) moduloone if for all intervals A ⊂ [0 ,
1) the asymptotic equality(1) lim N →∞ N N X n =1 A ( y n ) = λ ( A )holds. Here A is the indicator function of A , extended periodically with period 1, and λ denotes Lebesgue measure. Uniform distribution theory has a long history, going back tothe seminal paper of Hermann Weyl [43]. For general background, see [15, 24]. Uniformdistribution of a sequence can be seen as a pseudo-randomness property, in the sense thata sequence ( Y n ) n ≥ of independent, identically distributed random variables having uniformdistribution on [0 ,
1) satisfies (1) almost surely as a consequence of the Glivenko–Cantellitheorem; thus a deterministic sequence ( y n ) n ≥ which is uniformly distributed mod 1 exhibitsthe same behavior as a typical realization of a random sequence.A sequence ( y n ) n ≥ is said to have Poissonian pair correlation if for all real numbers s ≥ N →∞ N X ≤ m,n ≤ N,m = n [ − s/N,s/N ] ( y n − y m ) = 2 s. This notion is motivated by questions from theoretical physics, and plays a key role in theBerry–Tabor conjecture; see [27] for more information. Just like equidistribution, Poissonianpair correlation can also be seen as a pseudo-randomness property, since a random sequence( Y n ) n ≥ as above almost surely has Poissonian pair correlation. However, clearly the two Mathematics Subject Classification.
Primary 11K06, 11J83, 11M06; Secondary 11B05, 11J25, 11J71 .
Key words and phrases.
Pair correlation, Riemann zeta function, lattice points, Diophantine inequality. properties are of a rather different nature. While equidistribution is a “large-scale” statistic(where the test interval always remains the same), the pair correlation is a highly localizedstatistic (where the size of the test interval shrinks in proportion with N ). Note that the twoproperties are not independent: it is known that a sequence having Poissonian pair correlationnecessarily must be equidistributed [3, 17, 28], whereas the opposite implication is generallyfalse. An illustrative example is the sequence ( nα ) n ≥ , which is equidistributed if and only if α Q , but which fails to have Poissonian pair correlation for any α .The theory of uniform distribution modulo one can be said to be relatively well understood(at least in the one-dimensional case). Many specific sequences are known which are uni-formly distributed mod one. In contrast, only very few specific results are known in the paircorrelation setting. A notable exception is the sequence ( √ n ) n ∈ Z ≥ \ (cid:3) , which is known to havePoissonian pair correlation [16]. The sequence ( n α ) n ≥ is conjectured to have Poissonian paircorrelation under mild Diophantine assumptions on α , but only partial results are known inthis direction [20, 29, 33, 42]. Lacking specific examples, it is natural to turn to a metric the-ory instead. Let ( a n ) n ≥ be a sequence of distinct integers, let α ∈ R , and consider sequencesof the form ( a n α ) n ≥ . The metric theory of such sequences with respect to equidistributionis very simple: for every such ( a n ) n , the sequence ( a n α ) n is uniformly distributed mod 1for almost all α [43]. The situation with respect to pair correlation is much more delicate.Pioneering work in this area was carried out by Rudnick, Sarnak and Zaharescu [32, 35]. Asnoted above, ( nα ) n ≥ does not have Poissonian pair correlation for any α . However, for anypolynomial p ∈ Z [ X ] of degree at least 2, the pair correlation of ( p ( n ) α ) n is Poissonian foralmost all α . For related results, see for example [6, 12, 36].Recently, a simple criterion was established in [5] which allows to decide whether the sequence( a n α ) n has Poissonian pair correlation for almost all α for many naturally arising integersequences ( a n ) n . Let E N denote the number of solutions ( n , n , n , n ) of the equation(2) a n − a n + a n − a n = 0 , subject to 1 ≤ n , n , n , n ≤ N . This quantity is called the additive energy in the additivecombinatorics literature (see [18, 39]). Note that N ≤ E N ≤ N for every ( a n ) n and every N . The criterion is as follows. If a sequence ( a n ) n satisfies E N ≪ N − ε for some ε >
0, then( a n α ) n has Poissonian pair correlation for almost all α . If in contrast E N ≫ N , then theconclusion fails to be true. For further refinements of this criterion, and for remaining openproblems, see [4, 7, 8, 25]. We emphasize that all that was written in this paragraph requires( a n ) n to be a sequence of integers .Very little is known in the metric theory of pair correlation of sequences ( x n α ) n when ( x n ) n is asequence of reals rather than integers. One step in this general direction is [12], where ( x n ) n isallowed to take rational values and the results obtained depend on the size of the denominatorsof these rationals. A general result was obtained recently in [34], where the authors gave acriterion formulated in terms of the number of solutions of a certain Diophantine inequality.The criterion is as follows: for a sequence ( x n ) n , assume that there exist some ε > δ > n , n , n , n , j , j ) of the equation(3) | j ( x n − x n ) − j ( x n − x n ) | < N ε , PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 3 subject to 1 ≤ | j | , | j | ≤ N ε , ≤ n , n , n , n ≤ N, n = n , n = n , is of order ≪ N − δ , then ( x n α ) n has Poissonian pair correlation for almost all α . It is verified in [34]that this condition is satisfied for lacunary sequences. A condition in the spirit of (3) arisesvery naturally when studying this sort of problem (cf. also [35]); we will encounter a variantof this condition in Equation (14) below. In particular, it is very natural that in the integercase one has to count solutions of Diophantine equations , while in the real-number settingone has to count solutions of Diophantine inequations . The problem with (3) is that it isin general rather difficult to verify whether this condition is satisfied for a given sequence ornot, with issues being caused in particular by the presence of the coefficients j and j . Thepurpose of the present paper is to give a simplified criterion, in the spirit of the criterion of[5] which was specified in terms of the number of solutions of the equation (2). Theorem 1.
Let ( x n ) n ≥ be a sequence of positive real numbers for which there exists c > such that x n +1 − x n ≥ c, n ≥ . Let E ∗ N denote the number of solutions ( n , n , n , n ) of theinequality (4) | x n − x n + x n − x n | < , subject to n i ≤ N, i = 1 , , , . Assume that there exists some δ > such that E ∗ N ≪ N / − δ as N → ∞ . Then the sequence ( x n α ) n ≥ has Poissonian pair correlation for al-most all α ∈ R . The exponent 183 / ≈ .
408 in the conclusion of the theorem comes from a bound for the178 / E ∗ N can be relaxed to E ∗ N ≪ N − ε for any ε >
0, which would be in accordance with theresults known for the integer case.Theorem 1 applies, for example, to all sequences of the form x n = p ( n ) , n ≥
1, where p is aquadratic polynomial with real coefficients. For such a sequence ( x n ) n we have E ∗ N ≪ N ε for any ε > x n = p ( n ) for every polyno-mial p ∈ R [ X ] of degree d ≥
3, under the additional assumption that the coefficient of x d − is rational ; the required bound for E ∗ N then follows, after eliminating this coefficient, fromLemma 3 below (with the choice of θ = d and γ = N d − ). The extra assumption on the sec-ond coefficient is most likely redundant, but we have not been able to establish the necessarybound for E ∗ N without it. A famous open conjecture in additive combinatorics asserts that E N ≪ N ε for all convex sequences ( x n ) n , which would provide many further applicationsof our theorem; however, unfortunately the best current bound in this direction (Shkredov’s32 / ≈ .
46 from [38]) is just beyond the range of applicability of our theorem.Another particularly interesting case is when x n = n θ for some θ >
1. One then has E ∗ N ≪ N max { − θ + ε, ε } for any fixed ε > n θ α ) n ≥ has Poissonian pair correlationfor almost all α when θ > / ≈ . E ∗ N (see Lemma 3 below). This could be relaxed to assuming some Diophantine condition on this coefficient.
C. AISTLEITNER, D. EL-BAZ, M. MUNSCH
Using this additional information, we can use a modified version of the argument leading toTheorem 1, and prove that ( n θ α ) n ≥ actually has Poissonian pair correlation for all θ > Theorem 2.
For every real number θ > and almost every α ∈ R , the sequence ( n θ α ) n ≥ has Poissonian pair correlation. As noted above, the conclusion of Theorem 2 is not true when θ = 1. It seems plausible thatthe conclusion of the theorem is valid again for 0 < θ <
1. However, this cannot be provedwith the methods used in the present paper, which break down in the case of a sequence ( x n ) n whose order of growth is only linear or even slower. We will address this aspect at the veryend of the paper, where we also formulate some further open problems.In conclusion we note that Technau and Yesha recently obtained a result which is somewhatsimilar to our Theorem 2, but which is “metric” in the exponent rather than in a multiplicativeparameter. More precisely, they showed that ( n θ ) n has Poissonian pair correlation for almostall θ >
7. Their paper also contains similar results on higher correlations, which require alarger lower bound for θ . From a technical perspective, their problem is rather different fromours. For details see [41]. 2. Preliminaries
As in the introduction, let [ − s/N,s/N ] ( x ) denote the indicator function of the interval [ − s/N, s/N ],extended with period 1. That is, [ − s/N,s/N ] ( x ) = (cid:26) x − h x i ∈ [ − s/N, s/N ] , h x i denotes the nearest integer to x . We wish to show that under the assumptions ofTheorem 1 we have for almost all α ∈ R (5) 1 N X ≤ m,n ≤ N,m = n [ − s/N,s/N ] ( x m α − x n α ) → s as N → ∞ for all s ≥
0. It is well-known that for any s and N , and for any positive integer K there exist trigonometric polynomials f + K,s,N ( x ) and f − K,s,N ( x ) of degree at most K suchthat f − K,s,N ( x ) ≤ [ − s/N,s/N ] ( x ) ≤ f + K,s,N ( x )for all x , and such that Z f ± K,s,N ( x ) dx = 2 s/N ± K + 1 . Furthermore, the j -th Fourier coefficient c j of f − K,s,N satisfies(6) | c j | ≤ min (cid:18) sN , π | j | (cid:19) + 1 K + 1for all j , and an analogous bound holds for the Fourier coefficients of f + K,s,N . These trigono-metric polynomials are called Selberg polynomials, and their construction is described indetail in Chapter 1 of [30].
PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 5
Instead of establishing the required convergence relation (5) for indicator functions, we willrather work with the trigonometric polynomials f + K,s,N and f − K,s,N instead, which is technicallymore convenient. More precisely, in order to obtain (5) it suffices to prove the following. Forevery fixed positive integer r , and for every fixed real number s ≥
0, we have(7) 1 N X ≤ m,n ≤ N,m = n f + rN,s,N ( x m α − x n α ) ∼ N Z f + rN,s,N ( x ) dx as N → ∞ , for almost all α ∈ R , and the same is true when f + is replaced by f − . Toshow this convergence we prove that the expectation of the left-hand side of (7) with respectto α equals (asymptotically) the right-hand side of (7), and that the variance of the left-hand side of (7) is not too large. An application of Chebyshev’s inequality together with theBorel–Cantelli lemma then gives the desired result. As usual controlling the expectation iseasier than controlling the variance. We will obtain the required bound for the expectation inSection 3, and the bound for the variance in Sections 4 and 5. In Section 6 we conclude theproof of Theorem 1, and Section 7 contains all of the necessary modifications for the proof ofTheorem 2. 3. Proof of Theorem 1, Part 1: Controlling expectations
Let a positive integer r and a real number s ≥ f N for the function f + rN,s,N ,as defined in the previous section (or for the function f − rN,s,N — both cases work in exactlythe same way). We want to control the “expected value” with respect to α of the left-handside of (7) as N → ∞ . In the case when ( x n ) n ≥ is an integer sequence everything is periodicwith period 1, and it is appropriate to integrate over α ∈ [0 ,
1] with respect to the Lebesguemeasure. In our case, when ( x n ) n ≥ is a sequence of reals, we do not have such periodicity.We thus have to integrate over all α ∈ R with respect to an appropriate measure µ , which isabsolutely continuous with respect to the Lebesgue measure (so that a µ -almost everywhereconclusion implies a Lebesgue-almost everywhere conclusion). A good choice for the measure µ is the measure whose density with respect to the Lebesgue measure is given by(8) dµ ( x ) = 2(sin( x/ πx dx. The Fourier transform of x x/ πx is a non-negative real function which is supportedon the interval ( − , / √ π . Note that the measure µ is normalized such that µ ( R ) = 1.Expanding the function f N ( x ) = X j ∈ Z c j e πijx into a Fourier series, by construction we have c j = 0 when | j | > rN , and | c j | ≤ s/N +1 / ( rN ) ≪ N − (recall that r and s are assumed to be fixed). Moreover we have c = C. AISTLEITNER, D. EL-BAZ, M. MUNSCH R f N ( x ) dx . Using the fact that the Fourier transform of the measure µ is compactly sup-ported on ( − ,
1) and uniformly bounded, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N X ≤ m,n ≤ N,m = n f N ( x m α − x n α ) dµ ( α ) − N Z f N ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:18) N − N ( N − N (cid:19)| {z } =1 Z f N ( x ) dx | {z } ≪ N − + 1 N X ≤| j |≤ rN | c j | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R X ≤ m,n ≤ N,m = n e πij ( x m α − x n α ) dµ ( α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ N − + N − X ≤| j |≤ rN X ≤ m,n ≤ N,m = n (cid:0) | j ( x m − x n ) | < (cid:1)| {z } ≪ N due to the growth assumption on ( x n ) n ≥ ≪ N − , (9)where we estimated | c j | using (6). Thus the expected values behave as they should, since wehave(10) Z R N X ≤ m,n ≤ N,m = n f N ( x m α − x n α ) dµ ( α ) = N Z f N ( x ) dx + O (1 /N ) . Controlling the variances is more difficult, and will be done in the next two sections.4.
Proof of Theorem 1, Part 2: Controlling variances
In this section we keep the setup as in Section 3 above, that is, we assume that r and s are fixed, and we write f N for either f + rN,s,N or f − rN,s,N . Furthermore, we write h N for thecentered version of f N , that is, for the function(11) h N ( x ) = f N ( x ) − Z f ( x ) dx = X j ∈ Z ,j =0 c j e πijx . We wish to estimate the “variance” of our localized counting function, or more precisely thequantity(12) Var( h N , µ ) := Z R N X ≤ m,n ≤ N,m = n h N ( x m α − x n α ) dµ ( α ) . The following bound on Var( h N , µ ) is the crucial ingredient in our proof of Theorem 1. PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 7
Lemma 1.
For every ε > we have, as N → ∞ , (13) Var( h N , µ ) ≪ max (cid:16) N − ε/ + N − / ε ( E ∗ N ) / , E ∗ N N − . ε (cid:17) . For the convenience of the reader, we note at this point that our assumption that there issome δ > E ∗ N ≪ N / − δ ensures that there is some δ ′ > ε > h N , µ ) ≪ N − δ ′ + ε , which is sufficient to deduce Theorem 1 (seeSection 6 for details).We also note that under the Lindel¨of hypothesis, the bound that follows from our method isVar( h N , µ ) ≪ N − ε E ∗ N .5. Proof of Lemma 1: Lattice point counting via the Riemann zeta function
A first reduction.
Squaring out in (12) and using again the properties of the Fouriertransform of the measure µ , we can bound Var( h N , µ ) by Z R N X ≤ n ,n ,n ,n ≤ N,n = n , n = n X j ,j ∈ Z , j ,j =0 | j | , | j |≤ rN | c j c j | | {z } ≪ N − e πiα ( j ( x n − x n ) − j ( x n − x n )) dµ ( α ) ≪ N X ≤ n ,n ,n ,n ≤ N,n >n , n >n X ≤ j ,j ≤ rN (cid:0) | j ( x n − x n ) − j ( x n − x n ) | < (cid:1) , thereby essentially arriving at (3). For technical reasons, in this paper we prefer to localizethe variables j , j into dyadic regions and thus apply the Cauchy–Schwarz inequality to (12).To simplify later formulas we also replace the differences x n − x n and x n − x n by theirrespective absolute values using the parity of h N . Then, writing U for the smallest integerfor which 2 U ≥ rN , we have the following bound on the variance.Var( h N , µ ) ≪ Z R N U X u =1 X ≤ m,n ≤ N,m = n X u − ≤| j | < u c j e πij | x m − x n | α dµ ( α ) ≪ N Z R U X k =1 ! U X u =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ m,n ≤ N,m = n X u − ≤| j | < u c j e πij | x m − x n | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( α ) ≪ log NN U X u =1 X ≤ n ,n ,n ,n ≤ N,n = n , n = n X u − ≤ j ,j < u (cid:16)(cid:12)(cid:12)(cid:12) j | x n − x n | − j | x n − x n | (cid:12)(cid:12)(cid:12) < (cid:17) . (14)Thus we have reduced the problem of estimating the variance to a problem of bounding thenumber of solutions of a Diophantine inequation.5.2. Counting solutions by using the Riemann zeta function.
We will relate the count-ing problem in Equation (14) to the problem of bounding a twisted moment of the Riemannzeta function. Before we return to the proof, we point out the difference between the real-number case (in this paper) and the corresponding results for the case of ( x n ) n ≥ being an C. AISTLEITNER, D. EL-BAZ, M. MUNSCH integer sequence. In the integer case, the problem of estimating the variance of the pair cor-relation function can be reduced to counting solutions of j ( x n − x n ) = j ( x n − x n ). Notethat this is in accordance with the situation in the present paper, where we count solutions to | j ( x n − x n ) − j ( x n − x n ) | <
1, with the difference that in the integer case “ <
1” implies“= 0”. The number of solutions of the counting problem in the integer case is essentiallygoverned by what is called a “GCD sum”. It is known that such sums have a connection withthe Riemann zeta function (see [1, 21]), and strong estimates for such sums were obtained in[2, 9, 14]. Our argument below is motivated by a beautiful argument of Lewko and Radziwi l l[26], who showed how the relevant GCD sums can be estimated in terms of a twisted momentof a random model of the Riemann zeta function on the critical line. The randomizationwas crucial in their argument for different reasons, one being that the required distributionalestimates for extreme values of the actual Riemann zeta function are not known uncondition-ally. Their argument relied crucially on the fundamental theorem of arithmetic, and thus onthe fact that they were dealing with integer sequences. In the real-number case the situationis much more delicate. We will relate our counting problem to some convolution formula forthe Riemann zeta function. The kernel will be chosen for its good properties with respectto the Fourier transform (positivity and localized support) which allow to overcount withoutsubstantial loss. To summarize, in our argument below we will use of a combination of ideasfrom [1, 9, 10, 14] and [26].Let ( x n ) n ≥ be the sequence from the statement of Theorem 1. Let M = N − N , and let { z , . . . , z M } be the multi-set of all absolute differences {| x m − x n | : 1 ≤ m, n ≤ N, m = n } ,meaning that we allow repetitions in the definition. For a given positive integer u with2 u ≤ rN , we wish to estimate(15) X ≤ m,n ≤ M X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) . We write ζ ( σ + it ) for the Riemann zeta function. We also write Φ( t ) = e − t / , and note thatthis function has a positive Fourier transform given by b Φ = √ π Φ. Throughout the proof ε > N is “large”.Our argument proceeds by splitting into two cases depending on the size of min { z m , z n } . Wefirst treat the case when z m , z n are both at least of size N . . We then treat the case whenone of z m or z n is “small” which essentially amounts, because of our dyadic splitting, to thecase when both variables are small. • Case 1: Counting solutions for z m , z n ≥ N . . Let u be given such that 2 u − ≤ j , j ≤ u . Set T = 2 u N ε/ . For any integer k ≥
1, we set(16) b k = M X m =1 (cid:0) z m ∈ [ k, k + 1) (cid:1) . See also [13] for links between twisted moment of character sums and GCD sums. Furthermore, see [37]for a very recent paper of Shkredov, where he applies GCD sums and methods from [26] to give upper boundsfor the maximal length of arithmetic progressions contained in sets with small product set.
PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 9
Note that by definition the z m are non-negative, so ∞ X k =1 b k = M = N − N . We split theinterval [1 , ∞ ) into a disjoint union [ h ≥ I h , where I h = "&(cid:18) T (cid:19) h ' , &(cid:18) T (cid:19) h +1 '! , and we set(17) a h = X k ∈ I h b k / , h ≥ . Finally, we define a function(18) P ( t ) = ∞ X h =0 a h (cid:18) T (cid:19) iht . This function is constructed in such a way that Z R | P ( t ) | Φ( t/T ) dt = Z R X h ,h ≥ a h a h (cid:18) T (cid:19) ( h − h ) it Φ( t/T ) dt ≪ T X h ,h ≥ a h a h b Φ ( h − h ) ≪ T ∞ X h =0 a h (19)by Cauchy–Schwarz, due to the quick decay of b Φ. We will comment on the reasons for con-structing P ( t ) in this particular way in more detail at the end of our Case 1 analysis.Let j and j be fixed, and assume without loss of generality that j ≥ j . Let k be an integerin I h , and assume that z m ∈ [ k, k + 1). Then the inequality | j z m − j z n | < (cid:12)(cid:12)(cid:12)(cid:12)(cid:24) j kj (cid:25) − z n (cid:12)(cid:12)(cid:12)(cid:12) < j /j ≤ j , j are located in the same dyadic interval). We write ℓ ( k ) = ⌈ j k/j ⌉ . Recall that j /j ≥ k ℓ ( k ) is injective.Thus we have X z m ∈ I h ,z n ∈ I h (cid:0) | j z m − j z n | < (cid:1) ≪ X k ∈ I h X z m ∈ [ k,k +1) X z n ∈ I h , | ℓ ( k ) − z n | < ≪ X k ∈ I h X z m ∈ [ k,k +1) X − ≤ v ≤ X z n ∈ I h ,z n ∈ [ ℓ ( k )+ v,ℓ ( k )+ v +1) ≪ X − ≤ v ≤ X k ∈ I h such that ℓ ( k )+ v ∈ I h b k b ℓ ( k )+ v ≪ X k ∈ I h b k / X ℓ ∈ I h b ℓ / ≪ a h a h (21)by Cauchy–Schwarz.When j and j are fixed, there can only be solutions of | j z m − j z n | < z m ∈ I h and z n ∈ I h for particular pairs ( h , h ). Assume that z m ∈ I h and z n ∈ I h such that | j z m − j z n | <
1. Recall that j ≥ j by assumption, so we have (cid:12)(cid:12)(cid:12) z m z n − j j (cid:12)(cid:12)(cid:12) < j z n andconsequently z m z n ≤ j j + j z n ≤
2. Since z m ∈ I h and z n ∈ I h , the quotient z m /z n issomewhere between (1 + 1 /T ) h − h − and (1 + 1 /T ) h − h +1 , so that(22) z m z n ≤ (1 + 1 /T ) h − h | {z } ≤ /T ) ≤ , since z m /z n ≤ (1 + 1 /T ) ≤ (1 + 1 /T ) h − h + 3 T .
Similarly(23) z m z n ≥ (1 + 1 /T ) h − h − T .
Since j ≥ u − and z n ≥ N . by assumption, we have (cid:12)(cid:12)(cid:12) z m z n − j j (cid:12)(cid:12)(cid:12) ≤ u − N . ≤ T , wherethe last inequality follows from our choice of T . Overall, together with (22) and (23) thisshows that the inequality | j z m − j z n | < z m ∈ I h and z n ∈ I h is only possible when (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) T (cid:19) h − h − j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T .
Note that, for fixed j , j , this is an inequality which only depends on h , h and not on z m , z n anymore.Thus in combination with (21) we obtain X ≤ m,n ≤ M (cid:0) | j z m − j z n | < (cid:1) ≪ X h ,h ≥ , (cid:12)(cid:12)(cid:12) ( T ) h − h − j j (cid:12)(cid:12)(cid:12) ≤ T a h a h for all fixed j and j , and accordingly X ≤ m,n ≤ M X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) ≪ u X u − ≤ j ,j ≤ u j j ) / X h ,h ≥ , (cid:12)(cid:12)(cid:12) ( T ) h − h − j j (cid:12)(cid:12)(cid:12) ≤ T a h a h . (24)The next step is to relate this sum to an integral over the Riemann zeta function. Instead ofusing a truncated expression for ζ (1 / it ) (or an approximate functional equation) we will PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 11 make use of a convolution formula to avoid additional analytic difficulties. In this way we cantake advantage of the analytic properties of the Fourier transform of the kernel (positivityand localized support). A similar idea was also fruitfully used in a paper of Bondarenko andSeip [10]. For our purpose we use the following version, which is Lemma 5.3 of [14].
Lemma 2.
Let σ ∈ ] −∞ , and let F be a holomorphic function in the strip y = ℑ z ∈ [ σ − , ,such that (25) sup σ − ≤ y ≤ | F ( x + iy ) | ≪ x + 1 . Then for all s = σ + it ∈ C , t = 0 , we have X k,ℓ > b F (log kℓ ) k s ℓ s = Z R ζ ( s + iu ) ζ ( s − iu ) F ( u ) du + 2 πζ (1 − it ) F ( is − i ) + 2 πζ (1 + 2 it ) F ( is − i ) . We introduce the function K ( u ) := sin ((1 + ε/ u log N ) πu (1 + ε/ N ) , whose Fourier transform is given by b K ( ξ ) = max (cid:18) − | ξ | ε/
4) log
N , (cid:19) . We note that K can be extended analytically and satisfies (25). Furthermore(26) | K ( t − i/ | ≪ N ε/ / ( t + 1) , | K ( − t − i/ | ≪ N ε/ / ( t + 1) . The function K is chosen in such a way that we have b K (log j j ) ≫ − rN )2(1+ ε/
4) log N ≫ ε and r ).By the properties of b K and Φ we have X u − ≤ j ,j < u j j ) / X h ≥ ,h ≥ , (cid:12)(cid:12)(cid:12) ( T ) h − h − j j (cid:12)(cid:12)(cid:12) ≤ T a h a h ≪ X j ,j ≥ b K (log j j )( j j ) / X h ,h ≥ a h a h Φ (cid:18) T log (cid:18) j j (1 + 1 /T ) h − h (cid:19)(cid:19) ≪ T Z R X j ,j ≥ b K (log j j )( j j ) / (cid:18) j j (cid:19) it | P ( t ) | Φ( t/T ) dt. (27)Here P ( t ) is the function that we defined in (18). Note that in all three lines of the displayedequation above, all terms in the summations are non-negative, because b K, Φ and b Φ are allnon-negative.
We define G ( t ) = X j ,j ≥ b K (log j j )( j j ) / (cid:18) j j (cid:19) it . By Lemma 2 we have Z R G ( t ) | P ( t ) | Φ( t/T ) dt = Int + Int + Int , where Int = Z R | P ( t ) | Φ( t/T ) Z R ζ (1 / it + iu ) ζ (1 / − it + iu ) K ( u ) du dt, Int = 2 π Z R ζ (1 − it ) K ( − t − i/ | P ( t ) | Φ( t/T ) dt, Int = 2 π Z R ζ (1 + 2 it ) K ( t − i/ | P ( t ) | Φ( t/T ) dt. We trivially have the pointwise bound(28) | P ( t ) | ≤ | P (0) | = X h ≥ a h ≪ ∞ X n =1 b n ! ≪ N , t ∈ R . Splitting the integral and using (26) together with the easy estimate | ζ (1 + it ) | ≪ log t , weobtain Int ≪ N ε/ N (cid:18) Z t ≥ log tt dt (cid:19) ≪ N ε/ . Exactly the same estimate holds for Int .The classical convexity bound | ζ (1 / it ) | ≪ | t | / gives | ζ (1 / it + iu ) ζ (1 / − it + iu ) | ≪ ( | t | + | u | ) / ≪ | t | / + | u | / . Hence we can bound the contribution of the domain | u | ≥ T toInt by Z R | t | / | P ( t ) | Φ( t/T ) Z | u |≥ T K ( u ) du | {z } ≪ T − dt + Z R | P ( t ) | Φ( t/T ) Z | u |≥ T | u | / K ( u ) du | {z } ≪ T − / dt ≪ Z R | t | / T + 1 T / ! | P ( t ) | Φ( t/T ) dt ≪ T / ∞ X h =0 a h , where we used K ( u ) ≪ u − and (19) together with the quick decay of Φ.Define A = , so that by Ivi´c’s theorem [23, Theorem 8.3] we know(29) Z T | ζ (1 / it ) | A ≤ T M ( A )+ ε , where M ( A ) = 2 + A − = (and the first equality is valid for every A ∈ (cid:2) , (cid:3) ). PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 13
The contribution to Int of small u is Z | u |≤ T K ( u ) (cid:18)Z R | ζ (1 / it + iu ) || ζ (1 / − it + iu ) || P ( t ) | Φ( t/T ) dt (cid:19) du. To estimate the term inside the brackets, we use H¨older’s inequality with parameters 1 /A +1 /A + 1 /B = 1 with B = AA − = and write | P ( t ) | = | P ( t ) | − /B | P ( t ) | /B . By (19), (28)and (29) we deduce Z R | ζ (1 / it + iu ) || ζ (1 / − it + iu ) || P ( t ) | Φ( t/T ) dt (30) ≪ (cid:18)Z R | ζ (1 / it + iu ) | A Φ( t/T ) dt (cid:19) /A (cid:18)Z R | ζ (1 / − it + iu ) | A Φ( t/T ) dt (cid:19) /A ×× | P (0) | − /B ) (cid:18)Z R | P ( t ) | Φ( t/T ) dt (cid:19) /B ≪ (cid:16) T M ( A )+ ε (cid:17) /A N − /B ) T /B ∞ X h =0 a h ! /B = (cid:16) T M ( A )+ ε (cid:17) /A N /A T ( A − /A ∞ X h =0 a h ! ( A − /A . (31)Note that by the definition of a h in (17) we have ∞ X h =0 a h ! = ∞ X k =1 b k ≪ M X m,n =1 | z m − z n | < E ∗ N . Integrating over u we deduce thatInt ≪ (cid:16) T M ( A )+ ε (cid:17) /A N /A T ( A − /A ( E ∗ N ) ( A − /A (32) = T / ε N / ( E ∗ N ) / . (33)Using (24), (27) and inserting our bounds for Int , Int and Int we obtain X ≤ m,n ≤ M X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) ≪ u T Z R G ( t ) | P ( t ) | Φ( t/T ) dt ≪ u T (cid:16) N ε/ + T / ε N / ( E ∗ N ) / + T / E ∗ N (cid:17) ≪ u T (cid:16) N ε/ + T / ε N / ( E ∗ N ) / (cid:17) , where we used that the term T / E ∗ N is dominated by the other two summands. Substituting T = 2 u N ε/ and using that 2 u ≪ N , we finally get the upper bound X ≤ m,n ≤ M X u − ≤ j ,j < u ( | j z m − j z n | < ≪ N ε/ ( N ε/ + N / N × / ε ( E ∗ N ) / )= N − ε/ + N / ε ( E ∗ N ) / (34)We note here that conditionally under the Lindel¨of hypothesis we could estimate the integralin line (30) much more efficiently, by using a pointwise bound for the zeta function and esti-mating the remaining integral with (19).Before we move on to Case 2, we make some comments on our argument above. Intuitively, itwould seem more natural to work with Q ( t ) := P m z itm rather than with the more complicatedfunction P . However, from a technical point of view the key problem in the whole argument isto be able to choose an appropriate value of T which balances the contribution to our integralsof those values of t for with | t | is “large” (this gets worse when T is larger, since the boundfor the zeta function grows polynomially in T ) against the contribution coming from those t for which | t | is “small” (this contribution can only be compensated in the final estimate when T is sufficiently large). When working directly with Q , the size of T would need to depend onthe size of the z m in order to be able to control R | Q | . The “orthogonalization” procedureleading to our definition of P ( t ) gives us more freedom in our choice of T . The whole problemdiscussed in this paragraph occurs only in the real-number setting, in contrast to the integersetting. • Case 2: Counting solutions for z m , z n with min { z m , z n } ≤ N . . First consider the contribution to (15) of those z m and z n for which max { z m , z n } < N / .We assumed that x n +1 − x n ≥ c >
0, so z n ≥ c for all n . Furthermore, we deduce thatamong z , . . . , z M there are at most ≪ N / many elements which are smaller than 4 N / (we suppress the dependence of the implied constant on c ). Note that whenever j and z m , z n are fixed, there are at most ≪ j such that | j z m − j z n | < z n ≥ c . Thus the total contribution of pairs z m , z n with max { z m , z n } < N / toour counting problem is at most X ≤ m,n ≤ M max { z m ,z n } < N / X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) ≪ (cid:16) N / (cid:17) u ≪ N / . Now consider the case when max { z m , z n } ≥ N / . Recall that we have localized j , j intoa dyadic interval in the counting problem. This implies a similar localization for z m and z n ,since j /j ∈ [1 / ,
2] and | j z m − j z n | < z m /z n ∈ [1 / , { z m , z n } ≥ N / .Thus we can restrict ourselves in the counting problem (15) to the case when z m ∈ [4 N β , N β )for some β ≥ /
4, and when consequently z n needs to be in [ N β , N β ). Note that there are ≪ log N many intervals of this form necessary to cover the whole relevant range [ N / , N . ],and clearly we only need to consider 1 / ≤ β ≤ .
01. We count more solutions if we relaxthe condition to z m , z n ∈ [ N β , N β ). Thus, let us consider(35) X ≤ m,n ≤ M,z m ,z n ∈ [ N β , N β ) X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 15 for some β ∈ [1 / , . T = 2 u N β .We define the b k ’s as before but restricting ourselves to those z m contained in [ N β , N β ).That is, we set b k = X ≤ m ≤ M,z m ∈ [ N β , N β ) ( z m ∈ [ k, k + 1)) . Note that previously we had P k b k = M ≤ N , whereas now we have a stronger bound.Applying the Cauchy–Schwarz inequality we obtain(36) X k b k ≪ p E ∗ N N β/ . We define ( a h ) h ≥ and P ( t ) as in Case 1 (see (17) and (18)). In the present case the inequality | j z m − j z n | < (cid:12)(cid:12)(cid:12) z m z n − j j (cid:12)(cid:12)(cid:12) ≤ u − N β . By construction, 2 u − N β becomeslarge in comparison with T , and we can continue to argue as in Case 1. Note that now, asa consequence of (36), we have | P (0) | ≪ E ∗ N N β instead of | P (0) | ≪ N as in Case 1.Proceeding as in Case 1 we obtainInt , Int ≪ N β + ε/ E ∗ N + T / E ∗ N . As for Int , we now obtain, with the same notation as in Case 1 (meaning A = and M ( A ) = 29 / ≪ ( T M ( A )+ ε ) /A ( E ∗ N N β ) /A ( E ∗ N ) − /A T − /A (37) = E ∗ N ( T M ( A )+ ε ) /A ( N β ) /A T − /A . (38)Substituting T = 2 u N β ≪ N β , we conclude that (35) is bounded by ≪ u T (cid:16) E ∗ N N β + ε/ + E ∗ N T / ε N β/ (cid:17) (39) ≪ E ∗ N N ε/ + E ∗ N N / β/ β ) ε . (40)Recall that we only need to consider β ≤ .
01, and that there are at most ≪ log N manydifferent values of β to consider for Case 2. Hence it follows that, for every fixed value of u ,(35) is bounded by(41) ≪ E ∗ N N . ε . Let us remark that we could have used a different value of A here (for instance A = 12 forwhich M ( A ) = 2, namely Heath-Brown’s bound on the twelfth moment from [19]) to arriveat (41). However, we kept the same parameters as in Case 1 to simplify the writing.Finally, inserting in (14) the bound (34) from Case 1 together with the bound (41) yieldsVar( h N , µ ) ≪ max (cid:16) N − ε/ + N − / ε ( E ∗ N ) / , E ∗ N N − . ε (cid:17) , which concludes the proof of Lemma 1. Proof of Theorem 1: conclusion of the proof
The crucial ingredient in the proof of Theorem 1 is the variance bound from Lemma 1. Werecord that we have, for every ε > h N , µ ) ≪ max (cid:16) N − ε/ + N − / ε ( E ∗ N ) / , E ∗ N N − . ε (cid:17) . Inserting E ∗ N ≪ N / − δ shows that there is a small (fixed) constant δ ′ > ε > h N , µ ) ≪ N − δ ′ + ε . We can take for example ε = δ ′ /
2, so that (42) becomes Var( h N , µ ) ≪ N − δ ′ / . Everythingelse now follows from a standard procedure. To be a bit more specific, convergence in (7) canbe established using the estimate for the expectations in Section 3, and using the variancebound (42) together with Chebyshev’s inequality and the Borel–Cantelli lemma. From thatwe get a convergence result for almost all α ∈ R , for fixed values of r and s . One notes thatthere are only countably many possible values of r , and that by continuity/monotonicity it issufficient to consider countably many values of s . Since a countable union of sets of measurezero has measure zero as well, almost all α ∈ R have the property that (7) holds for all r andall s , as desired. We refer the reader to [5] or [34], where this argument is carried out in fulldetail. It applies without any modifications to the situation in the present paper.7. Proof of Theorem 2.
Throughout this section we assume that θ > x n )defined by x n = n θ , n ≥
1. Note that with this definition we have, for every n ≥ , x n +1 − x n ≥
1, so the assumption x n +1 − x n ≥ c of Theorem 1 is satisfied in this case with c = 1. For ourproof of Theorem 2 we will rely on the following lemma of Robert and Sargos. Lemma 3 ([31, Theorem 2]) . Let θ = 0 , be a fixed real number. For any γ > and M ≥ ,let N ( M, γ ) denote the number of 4-tuples ( n , n , n , n ) ∈ { M + 1 , M + 2 , . . . , M } forwhich (43) (cid:12)(cid:12)(cid:12) n θ − n θ + n θ − n θ (cid:12)(cid:12)(cid:12) ≤ γ. Then for every ε > , N ( M, γ ) ≪ ε M ε + γM − θ + ε . The restriction to a dyadic range for ( n , n , n , n ) in the statement of the lemma doesnot actually play a role. This is easily seen by interpreting the number of solutions of theinequality as an L -norm. Indeed, generalizing the definition in (8) and setting dµ γ ( x ) = (sin( γx )) πγx dx, we have a measure whose Fourier transform is a (normalized) tent function on [ − γ, γ ]. Let E ∗ N,γ denote the number of solutions of (43), subject to ( n , n , n , n ) ∈ { , . . . , N } . Assume We thank Niclas Technau for pointing out to us that the estimate in Lemma 3 is also contained as a specialcase in a general result in a very recent paper of Huang [22]. Huang’s result gives improved error terms, butfor our application this does not play a role. However, the generality of Huang’s results could allow furtherapplications of our method in the spirit of our Theorem 2.
PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 17 for simplicity of writing that N is a power of 2, i.e. N = 2 L for some L ≥
1. Then applyingH¨older’s inequality we obtain E ∗ N,γ ≪ Z R L X ℓ =0 X ℓ −
0. Note that for θ ≥ γ = 1) that we can applyTheorem 1 and derive the desired conclusion. Thus in the sequel we can restrict ourselves tothe case θ < X ≤ m,n ≤ M X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) where now { z , . . . , z M } is the multi-set of all the absolute differences {| m θ − n θ | : 1 ≤ m, n ≤ N, m = n } . As above, u is a positive integer with 2 u ≤ rN , and M = N − N .As in the general argument before, we can easily dispose of the contribution of those z m , z n for which max { z m , z n } < N / . Thus again we can localize z m and z n , and restrict ourselvesto counting(46) X ≤ m,n ≤ M,z m ,z n ∈ [ N β , N β ) X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) , where in the present situation we only need to consider values of β in the range from 1 / θ . Note that at most ≪ log N many different values of β need to be considered to cover thewhole relevant range. Let β ∈ [1 / , θ ] be fixed. Let u in (46) be fixed. For integers k ≥ define(47) b k = X ≤ m ≤ M,z m ∈ [ N β , N β ) (cid:18) z m ∈ (cid:20) k u , k + 12 u (cid:19)(cid:19) . Note the difference in comparison with (16). There we collected all z m in a range of the form[ k, k + 1), since we could only control the number of solutions of the specific inequality (4),which has “ <
1” on the right-hand side. In contrast, by (43) we can now control the numberof solutions on a finer scale, and can accordingly set shorter ranges for the grouping of the z m (where γ = 2 − u ).Let ε > ε ≤ θ −
1. Set T = 2 u N min { β − ε, ε } . Unlike theargument in the general case in Section 4, we do not explicitly distinguish between Case 1and Case 2 (“large” and “small”), but have implicitly included this distinction into the waythat T is defined. As in Section 4, we split the interval [1 , ∞ ) into a disjoint union S ∞ h =0 I h ,where I h = "&(cid:18) T (cid:19) h ' , &(cid:18) T (cid:19) h +1 '! , and set a h = X k : k/ u ∈ I h b k / , h ≥ , as well as P ( t ) = ∞ X h =0 a h (cid:18) T (cid:19) iht . Then by construction we again have Z R | P ( t ) | Φ( t/T ) dt ≪ T ∞ X h =0 a h as in Section 4.From the particular structure and growth of the sequence ( n θ ) we can easily see that(48) X k b k ≤ n m ≤ M : z m ≤ N β o ≪ N β − θ , so that now we have | P (0) | ≪ N β − θ .As in Section 4, we assume without loss of generality that j ≥ j . Recall that j , j ≥ u − by assumption. Assume that z m ∈ [ k/ u , ( k + 1) / u ) for some k . Then the inequality | j z m − j z n | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l j kj m u − z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ u , PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 19 which is a version of (20) that is adapted to the construction in (47). Arguing as in the linesleading to (21), this again gives X z m ∈ I h ,z n ∈ I h (cid:0) | j z m − j z n | < (cid:1) ≪ a h a h , which perfectly resembles (21) but where now the b k and a h are defined in a different wayaccording to (47). Note that(49) X h ≥ a h ≪ X k ≥ b k ≪ E ∗ N, − u ≪ N ε + 2 − u N − θ + ε by (45). Note also that T is chosen in such a way that j z m and j z n exceed T ; indeed, byassumption j , j ≥ u − and z m , z n ≥ N β , while T ≤ u N β − ε by definition. Thus we cancontinue the argument as in Section 5. It turns out that in this setting it is sufficient to usethe Weyl–Hardy–Littlewood bound | ζ (1 / it ) | ≪ | t | / ε in order to bound Int , ratherthan the more elaborate argument relying on estimates for moments of the Riemann zetafunction. We obtain X ≤ m,n ≤ M,z m ,z n ∈ [ N β , N β ) X u − ≤ j ,j < u (cid:0) | j z m − j z n | < (cid:1) ≪ u T (cid:16) | P (0) | N ε/ + T / E ∗ N, − u (cid:17) ≪ u N β − θ N ε/ T + (2 u ) / N (1+2 ε ) / N ε + (2 u ) / N (1+2 ε ) / N − θ + ε ≪ N max { β − θ +5 ε/ , β − θ ) − ε/ } + N / ε + N / − θ +2 ε ≪ N − ε , where we used that β ≤ θ ≤ , θ ≥ ε, T ≪ u N ε , u ≪ N . Noting that we needto consider at most ≪ log N different values of β , this gives the necessary variance estimate.The remaining part of the proof of Theorem 2 can be carried out exactly as in the proof ofTheorem 1. We remark that any subconvex bound for the Riemann zeta function would besufficient to derive the same conclusion.8. Closing remarks
As remarked in the introduction, our method breaks down completely when the growth orderof the sequence ( x n ) n ≥ is only linear or even slower. Not only does the “lattice point countingwith the zeta function” argument from Section 5 fail to work in this situation, but there is amuch more fundamental reason why the whole approach based on calculating first and secondmoments (expectations and variances, as in Sections 3 and 4) fails to work in this setup. Togive a brief sketch of what causes the problem, assume that ( x n ) n is a sequence of reals suchthat x n ≤ n, n ≥
1. Assume that we want to bound the variance in analogy with (12), sosay we want to show that(50) Z R N X ≤ m,n ≤ N,m = n [ − /N, /N ] ( x m α − x n α ) dµ ( α ) tends to zero as N → ∞ (where we write the original indicator function instead of its approx-imation by a trigonometric polynomial, and where for simplicity of writing we set s = 1). Byour assumption on the growth of ( x n ) n , all differences x m − x n appearing in the sum aboveare uniformly bounded by N . Thus we have [ − /N, /N ] ( x m α − x n α ) = 1 throughout therange α ∈ [ − /N , /N ], for all m, n ≤ N . Consequently Z R N X ≤ m,n ≤ N,m = n [ − /N, /N ] ( x m α − x n α ) dµ ( α ) ≥ Z /N − /N N X ≤ m,n ≤ N,m = n dµ ( α ) ≫ . Thus the variance fails to tend to zero for a slowly growing ( x n ) n , due to the fact the contribu-tion of small values of α to the variance integral is too large. The argument used in Section 5fails to work in a similar way for slowly growing ( x n ) n , since the error terms coming from thecontribution to the integrals of values of t near zero become too large. Consequently, it seemsthat for establishing Poissonian pair correlation of ( x n α ) n for almost all α for slowly growing( x n ) n some genuine new ideas are necessary. Note that we cannot simply remove all valuesof α near zero from the variance integral (50) by replacing µ with some other measure whichvanishes for small α , since such a measure would fail to have non-negative Fourier transform(thereby causing major problems in other places). Note also that all these problems withslowly growing sequences ( x n ) n are a novel aspect which only shows up in the real-numbersetup – in contrast, when ( a n ) n is an integer sequence which grows at most linearly, then( a n α ) n is known to fail to have Poissonian pair correlation for any α , because the additiveenergy of ( a , . . . , a N ) necessarily is of maximal possible order (cf. [25]).We emphasize that the fact that our method fails to work in the case of slowly growingsequences ( x n ) n should not be understood as indicating that in such a case ( x n α ) n shouldnecessarily fail to have Poissonian pair correlation for almost all α . Quite on the contrary,there are good reasons to expect that also for slowly growing ( x n α ) n one should in “generic”situations obtain Poissonian pair correlation for almost all α . It seems that the propertyof having Poissonian pair correlation for ( x n α ) n for almost all α can only be prevented bya certain (“small-scale”) combinatorial obstruction, in such a way that the case of integersequences ( x n ) n with slowly growing ( x n ) n can be seen as a degenerate situation exhibitingexactly this type of combinatorial obstruction (coming from the fact that in the integer setup A similar argument appears at the end of [32], where it is used to show that the L approach fails to workin the case of the triple correlation of ( n α ) n ; cf. also [40] It might be difficult to spot at a quick glance, so we briefly comment on where the speed of growth of ( x n ) n was used in our argument in Sections 5 and 7. There is a term | P (0) | N ε/ coming from the contributionof values of t near the origin to the integral. This term is divided by T at the end of the calculation, so wecannot take T too small since we need N | P (0) | N ε/ /N T →
0. On the other hand, we cannot take T toolarge, since we need T ≪ u z n to be able to detect the solutions of our Diophantine inequality. To balanceeverything out, we need to be able to assure that there are not too many small values of z n (i.e., not too manydifferences x m − x n which are “small”). In our proof of Theorem 1 our assumption on the order of the additiveenergy takes care of this: it is easy to see that an upper bound on E ∗ N implies an upper bound on the numberof “small” differences x m − x n ; this is what we used in Case 2 of Section 5. In the setting of Theorem 2 wecan control the number of small differences x m − x n efficiently because of the particularly simple structure ofthe sequence. The relevant equations are (36) and (48). PAIR CORRELATION PROBLEM AND COUNTING LATTICE POINTS 21 everything which is smaller than one in absolute value necessarily equals zero). We believethat these are very interesting phenomena, and we propose the following open problems.
Open Problem 1:
Let θ ∈ (0 , n θ α ) n ≥ has Poissonian pair correlation foralmost all α . Note that Lemma 3 is still valid for this range of θ . Open Problem 2:
Let x n = n + log n . Show that ( x n α ) n has Poissonian pair correlationfor almost all α . We note that it is possible to establish a variant of Lemma 3 for this setting(with exponent 3 in place of 4 − θ ). Open Problem 3:
Let x n = n log n, n ≥
1. Show that ( x n α ) n has Poissonian pair correla-tion for almost all α .Clearly the exponent 183 / − δ in the statement of Theorem 1 is not optimal, and mostlikely it can be improved to 3 − δ (which is the case conditionally under the Lindel¨of hypoth-esis). It seems to us that the method of Bloom and Walker [8], which led to a quantitativeimprovement of the results of [5], cannot be used here. Their method relied on sum-productestimates, which, roughly speaking, leads to an integrand | P ( t ) | being replaced by | P ( t ) | .In the case of integer sequences (when working with the random model of the zeta function)one has perfect orthogonality, so that R | P | can be efficiently bounded. In our setting thesituation is quite different – we have constructed our function P ( t ) in such a way that thediagonal contribution dominates when calculating R | P | , but we do not have orthogonalityfor R | P | and cannot efficiently bound this integral. Open Problem 4:
Show that Theorem 1 remains valid under the weaker assumption E ∗ N ≪ N − δ for some δ >
0. Show that this can be further relaxed to assuming E ∗ N,γ ≪ γN − δ , forall γ in a range from roughly 1 /N to 1. It might even be the case that only values of γ neara critical size of roughly 1 /N are relevant.As noted, in the case of an integer sequence ( x n ) n it is known that ( x n α ) n cannot have Pois-sonian pair correlation for almost all α when E N ≫ N . It is unclear if there is a similarphenomenon in the real-number case. Open Problem 5:
Show that unlike in the integer case, it is possible for an increasingsequence ( x n ) n ≥ of reals that E ∗ N ≫ N and that ( x n α ) n has Poissonian pair correlation foralmost all α (compare Open Problems 1 and 2 above, where E ∗ N ≫ N ). Establish a criterion(stated for example in terms of E ∗ N,γ ) which ensures that ( x n α ) n does not have Poissonianpair correlation for almost all α . A candidate for such a criterion is that E ∗ N,γ ≫ γN forsome γ = γ ( N ) for infinitely many N , where maybe one also has to assume that these valuesof γ are of size γ ≈ /N . Acknowledgements
CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945 andY-901. DE is supported by FWF projects F-5512 and Y-901. MM is supported by FWFproject P-33043. We thank Winston Heap, Olivier Robert, Zeev Rudnick, Ilya Shkredov,Igor Shparlinski, Athanasios Sourmelidis and Niclas Technau for discussions and comments.
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