A pair correlation problem, and counting lattice points with the zeta function

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. The metric theory of pair correlations of sequences of the form $(a_n \alpha)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $\alpha$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $\alpha$, in terms of the additive energy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper we develop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $\theta>1$, the sequence $(n^\theta \alpha)_{n \geq 1}$ has Poissonian pair correlation for almost all $\alpha \in \mathbb{R}$.


Introduction and statement of results
A sequence (y n ) n≥1 of real numbers is called uniformly distributed (or equidistributed) modulo one if for all intervals A ⊂ [0, 1) the asymptotic equality (1) lim holds. Here 1 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 to the seminal paper of Hermann Weyl [42]. For general background, see [15,24]. Uniform distribution of a sequence can be seen as a pseudo-randomness property, in the sense that a sequence (Y n ) n≥1 of independent, identically distributed random variables having uniform distribution on [0, 1) satisfies (1) almost surely as a consequence of the Glivenko-Cantelli theorem; thus a deterministic sequence (y n ) n≥1 which is uniformly distributed mod 1 exhibits the same behavior as a typical realization of a random sequence.
A sequence (y n ) n≥1 is said to have Poissonian pair correlation if for all real numbers s ≥ 0, This notion is motivated by questions from theoretical physics, and plays a key role in the Berry-Tabor conjecture; see [27] for more information. Just like equidistribution, Poissonian pair correlation can also be seen as a pseudo-randomness property, since a random sequence (Y n ) n≥1 as above almost surely has Poissonian pair correlation. However, clearly the two 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 localized statistic (where the size of the test interval shrinks in proportion with N ). Note that the two properties are not independent: it is known that a sequence having Poissonian pair correlation necessarily must be equidistributed [3,17,28], whereas the opposite implication is generally false. An illustrative example is the sequence (nα) n≥1 , 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 uniformly distributed mod one. In contrast, only very few specific results are known in the pair correlation setting. A notable exception is the sequence ( √ n) n∈Z ≥1 \ , which is known to have Poissonian pair correlation [16]. The sequence (n 2 α) n≥1 is conjectured to have Poissonian pair correlation under mild Diophantine assumptions on α, but only partial results are known in this direction [20,29,33,41]. Lacking specific examples, it is natural to turn to a metric theory instead. Let (a n ) n≥1 be a sequence of distinct integers, let α ∈ R, and consider sequences of the form (a n α) n≥1 . The metric theory of such sequences with respect to equidistribution is very simple: for every such (a n ) n , the sequence (a n α) n is uniformly distributed mod 1 for almost all α [42]. 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]. As noted above, (nα) n≥1 does not have Poissonian pair correlation for any α. However, for any polynomial p ∈ Z[x] of degree at least 2, the pair correlation of (p(n)α) n is Poissonian for almost 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 integer sequences (a n ) n . Let E N denote the number of solutions (n 1 , n 2 , n 3 , n 4 ) of the equation (2) a n 1 − a n 2 + a n 3 − a n 4 = 0, subject to 1 ≤ n 1 , n 2 , n 3 , n 4 ≤ N . This quantity is called the additive energy in the additive combinatorics literature (see [18,38]). Note that N 2 ≤ E N ≤ N 3 for every (a n ) n and every N . The criterion is as follows. If a sequence (a n ) n satisfies E N ≪ N 3−ε for some ε > 0, then (a n α) n has Poissonian pair correlation for almost all α. If in contrast E N ≫ N 3 , then the conclusion fails to be true. For further refinements of this criterion, and for remaining open problems, 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 a sequence of reals rather than integers. One step in this general direction is [12], where (x n ) n is allowed to take rational values and the results obtained depend on the size of the denominators of these rationals. A general result was obtained recently in [34], where the authors gave a criterion 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 > 0 and δ > 0 such that the number of solutions (n 1 , n 2 , n 3 , n 4 , j 1 , j 2 ) of the equation subject to 1 ≤ |j 1 |, |j 2 | ≤ N 1+ε , 1 ≤ n 1 , n 2 , n 3 , n 4 ≤ N, n 1 = n 2 , n 3 = n 4 , is of order ≪ N 4−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) arises very naturally when studying this sort of problem (cf. also [35]); we will encounter a variant of this condition in Equation (12) below. In particular, it is very natural that in the integer case one has to count solutions of Diophantine equations, while in the real-number setting one has to count solutions of Diophantine inequations. The problem with (3) is that it is in general rather difficult to verify whether this condition is satisfied for a given sequence or not, with issues being caused in particular by the presence of the coefficients j 1 and j 2 . The purpose 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).
Let (x n ) n≥1 be a sequence of positive real numbers for which there exists c > 0 such that x n+1 − x n ≥ c, n ≥ 1. Let E * N denote the number of solutions (n 1 , n 2 , n 3 , n 4 ) of the inequality subject to n i ≤ N, i = 1, 2, 3, 4. Assume that there exists some δ > 0 such that E * N ≪ N 220/91−δ as N → ∞. Then the sequence (x n α) n≥1 has Poissonian pair correlation for almost all α ∈ R.
The exponent 220/91 ≈ 2.418 in the conclusion of the theorem comes from a bound for the 15-th moment of the Riemann zeta function on the critical line due to Heath-Brown [19], combined with Bourgain's [10] estimate |ζ(1/2 + it)| ≪ t 13/84+ε . Using only one of these two ingredients would lead to a slightly weaker exponent. Conditionally under the Lindelöf hypothesis, our bound for E * N can be relaxed to E * N ≪ N 3−ε for any ε > 0, which would be in accordance with the results 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 a quadratic polynomial with real coefficients. For such a sequence (x n ) n we have E * N ≪ N 2+ε for any ε > 0 by Lemma 5.2 of [11]. Theorem 1 also applies to x n = p(n) for every polynomial p ∈ R[X] of degree d ≥ 3, under the additional assumption that the coefficient of x d−1 is rational 1 ; the required bound for E * N then follows, after eliminating this coefficient, from Lemma 2 below (with the choice of θ = d and γ = N d−2 ). The extra assumption on the second coefficient is most likely redundant, but we have not been able to establish the necessary bound for E * N without it. A famous open conjecture in additive combinatorics asserts that E N ≪ N 2+ε for all convex sequences (x n ) n , which would provide many further applications of our theorem; however, unfortunately the best current bound (Shkredov's 32/13 ≈ 2.46 [37]) is just beyond the range of applicability our theorem.
Another particularly interesting case is when x n = n θ for some θ > 1. One then has E * N ≪ N max{4−θ+ε,2+ε} for any fixed ε > 0, so that as a direct application of Theorem 1 we can immediately deduce that (n θ α) n≥1 has Poissonian pair correlation for almost all α when θ > 144/91 ≈ 1.582. However, for this particular sequence we have additional information supplementing our knowledge of the order of E * N (see Lemma 2 below). Using this additional information, we can use a modified version of the argument leading to Theorem 1, and prove that (n θ α) n≥1 actually has Poissonian pair correlation for all θ > 1.
As noted above, the conclusion of Theorem 2 is not true when θ = 1. It seems plausible that the conclusion of the theorem is valid again for 0 < θ < 1. However, this cannot be proved with 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 very end 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 somewhat similar to our Theorem 2, but which is "metric" in the exponent rather than in a multiplicative parameter. More precisely, they showed that (n θ ) n has Poissonian pair correlation for almost all θ > 7. Their paper also contains similar results on higher correlations, which require a larger lower bound for θ. From a technical perspective, their problem is rather different from ours. For details see [40]. 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 such that

Preliminaries
for all j, and an analogous bound holds for the Fourier coefficients of f + K,s,N . These trigonometric polynomials are called Selberg polynomials, and their construction is described in detail in Chapter 1 of [30].
Instead of establishing the required convergence relation (5) for indicator functions, we will rather work with the trigonometric polynomials f + K,s,N and f − K,s,N instead, which is technically more convenient. More precisely, in order to obtain (5) it suffices to prove the following. For every fixed positive integer r, and for every fixed real number s ≥ 0, we have as N → ∞, for almost all α ∈ R, and the same is true when f + is replaced by f − . To show this convergence we prove that the expectation of the left-hand side of (6) with respect to α equals (asymptotically) the right-hand side of (6), and that the variance of the lefthand side of (6) is not too large. An application of Chebyshev's inequality together with the Borel-Cantelli lemma then gives the desired result. As usual controlling the expectation is easier than controlling the variance. We will obtain the required bound for the expectation in Section 3, and the bound for the variance in Sections 4 and 5. In Section 6 we conclude the proof of Theorem 1, and Section 7 contains all of the necessary modifications for the proof of Theorem 2.

Proof of Theorem 1, Part 1: Controlling expectations
Let a positive integer r and a real number s ≥ 0 be fixed. Write 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 exactly the same way). We want to control the "expected value" with respect to α of the left-hand side of (6) as N → ∞. In the case when (x n ) n≥1 is an integer sequence everything is periodic with period 1, and it is appropriate to integrate over α ∈ [0, 1] with respect to the Lebesgue measure. In our case, when (x n ) n≥1 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 is absolutely continuous with respect to the Lebesgue measure (so that a µ-almost everywhere conclusion 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 (7) dµ(x) = 2(sin(x/2)) 2 πx 2 dx.
The Fourier transform of 2(sin(x/2)) 2 πx 2 is a non-negative real function which is supported on the interval (−1, 1), and which is uniformly bounded by 1/ √ 2π. Note that the measure µ is normalized such that µ(R) = 1.
Expanding the function into a Fourier series, by construction we have c j = 0 when |j| > rN , and |c j | ≤ 2s/N + 1/(rN ) ≪ N −1 (recall that r and s are assumed to be fixed). Moreover we have c 0 = 1 0 f N (x)dx. Using the fact that the Fourier transform of the measure µ is compactly supported on (−1, 1) and uniformly bounded, we obtain Thus the expected values behave as they should, since we have Controlling the variances is more difficult, and will be done in the next two sections.

Proof of Theorem 1, Part 2: Controlling variances
In this section we will keep the setup as in Section 3 above, that is, we assume that r and s are fixed, and we write f N for f + rN,s,N or f − rN,s,N . Furthermore, we write h N for the centered version of f N , that is, for the function (10) h We wish to estimate the "variance" of our localized counting function, precisely Squaring out and using again the properties of the Fourier transform of the measure µ, we can bound this variance by thereby essentially arriving at (3). For technical reasons, in this paper we prefer to localize the variables j 1 , j 2 into dyadic regions and thus apply the Cauchy-Schwarz inequality to (11).
To simplify later formulas we also replace the differences x n 1 − x n 2 and x n 3 − x n 4 by their respective absolute values using the parity of h N . Then, writing U for the smallest integer for which 2 U ≥ rN , the variance defined in (11) is bounded by Thus we have reduced the problem of estimating the variance to a problem of bounding the number of solutions of a Diophantine inequation. We will estimate this number of solutions in the next section.

Proof of Theorem 1, Part 3: Lattice point counting via the Riemann zeta function
We will relate the counting problem in Equation (12) to the problem of bounding a twisted moment of the Riemann zeta function. Before we turn 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≥1 being an integer sequence. In the integer case, the problem of estimating the variance of the pair correlation function can be reduced to counting solutions of . Note that is in accordance with the situation in the present paper, where we count solutions to |j 1 (x n 1 − x n 2 )− j 2 (x n 3 − x n 4 )| < 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 essentially governed by what is called a "GCD sum". It is known that such sums have a connection with the 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 moment of a random model of the Riemann zeta function on the critical line. 2 The randomization was crucial in their argument for different reasons, one being that the required distributional estimates for extreme values of the actual Riemann zeta function are not known unconditionally. Their argument relied crucially on the fundamental theorem of arithmetic, and thus on the fact that they were dealing with integer sequences. In the real-number case the situation is much more delicate. In our argument below we will use a combination of ideas from [1,9,14] and [26].
Let (x n ) n≥1 be the sequence from the statement of Theorem 1. Let M = N 2 − N , and let {z 1 , . . . , 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 u ≤ 2rN , we wish to estimate (13) 1≤m,n≤M, m =n We write ζ(σ + it) for the Riemann zeta function. We also write Φ(t) = e −t 2 /2 , and note that this function has a positive Fourier transform given by Φ = √ 2πΦ. Throughout the proof ε > 0 is a small constant, and we assume that N is "large".
We split the argument into two cases depending on the size of min{z m , z n }. First we treat the case when z m , z n are both at least of size N 1.01 . After that we treat the case when both z m and z n are "small".
• Case 1: Counting solutions for z m , z n ≥ N 1.01 .
Let u be given such that 2 u−1 ≤ j 1 , j 2 ≤ 2 u . Set T = 2 u N 1+ε/2 . For any integer k ≥ 0, we set (14) b Note that by definition the z m are non-negative, so Finally, we define a function This function is constructed in such a way that by Cauchy-Schwarz, due to the quick decay of Φ.
Let j 1 and j 2 be fixed, and assume w.l.o.g. that j 1 ≥ j 2 . Let k be an integer in I h 1 , and assume that z m ∈ [k, k + 1). Then the inequality |j 1 z m − j 2 z n | < 1 is only possible when (recall that j 1 /j 2 ≤ 2 because j 1 , j 2 are located in the same dyadic interval). We write ℓ(k) = ⌈j 1 k/j 2 ⌉. Recall that j 1 /j 2 ≥ 1 by assumption, so the mapping k → ℓ(k) is injective. Thus we have by Cauchy-Schwarz.
When j 1 and j 2 are fixed, there can only be solutions of |j 1 z m − j 2 z n | < 1 with z m ∈ I h 1 and z n ∈ I h 2 for particular pairs (h 1 , h 2 ). Assume that z m ∈ I h 1 and z n ∈ I h 2 such that |j 1 z m − j 2 z n | < 1. Recall that j 1 ≥ j 2 by assumption, so we have zm zn − j 2 j 1 < 1 j 1 zn and consequently zm zn ≤ j 2 j 1 + 1 j 1 zn ≤ 2. Since z m ∈ I h 1 and z n ∈ I h 2 , the quotient z m /z n is somewhere between ( Since j 1 ≥ 2 u−1 and z n ≥ N 1.01 by assumption, we have zm zn − j 2 j 1 ≤ 1 2 u−1 N 1.01 ≤ 1 T , where the last inequality follows from our choice of T . Overall, together with (20) and (21) this shows that the inequality |j 1 z m − j 2 z n | < 1 for z m ∈ I h 1 and z n ∈ I h 2 is only possible when Note that, for fixed j 1 , j 2 , this is an inequality which only depends on h 1 , h 2 and not on z m , z n anymore. Thus in combination with (19) we obtain 1≤m,n≤M m =n for all fixed j 1 and j 2 , and accordingly 1≤m,n≤M m =n The next step is to relate this sum to an integral over the Riemann zeta function.
The following Lemma is Lemma 5.3 of [14]. |F (x + iy)| ≪ 1 Then for all s = σ + it ∈ C, t = 0, we have We introduce the function whose Fourier transform is given by We note that K can be extended analytically and satisfies (23). Furthermore (24) The function K is chosen in such a way that we have K(log j 1 j 2 ) ≫ 1 − 2(log rN ) 2(1+ε/4) log N ≫ 1 (where we suppress the dependence on the constants ε and r).
By the properties of K and Φ we have Here P (t) is the function that we defined in (16). Note that in all three lines of the displayed equation above, all terms in the summations are non-negative, because K, Φ and Φ are all non-negative.
We note here that conditionally under the Lindelöf hypothesis we could estimate the integral in line (29) much more efficiently, by using a pointwise bound for the zeta function and estimating the remaining integral with (17).
• Case 2: Counting solutions for z m , z n with min{z m , z n } ≤ N 1.01 .
First consider the contribution to (13) of those z m and z n for which max{z m , z n } < 4N 1/4 . We assumed that x n+1 − x n ≥ c > 0, so z n ≥ c for all n. Furthermore, we deduce that among z 1 , . . . , z M there are at most ≪ N 5/4 many elements which are smaller than 4N 1/4 (we suppress the dependence of the implied constant on c). Note that whenever j 1 and z m , z n are fixed, there are at most ≪ 1 many possible choices for j 2 such that |j 1 z m − j 2 z n | < 1, again since z n ≥ c. Thus the total contribution of pairs z m , z n with max{z m , z n } < 4N 1/4 to our counting problem is at most 1≤m,n≤M max{zm,zn}<4N 1/4 Now consider the case when max{z m , z n } ≥ 4N 1/4 . Recall that we have localized j 1 , j 2 into a dyadic interval in the counting problem. This implies a similar localization for z m and z n , since j 1 /j 2 ∈ [1/2, 2] and |j 1 z m − j 2 z n | < 1 are only possible if we have z m /z n ∈ [1/4, 4], given the fact that max{z m , z n } ≥ 4N 1/4 .
Thus we can restrict ourselves in the counting problem (13) to the case when z m ∈ [4N β , 8N β ) for some β ≥ 1/4, and when consequently z n needs to be in [N β , 32N β ). Note that there are ≪ log N many intervals of this form necessary to cover the whole relevant range [N 1/4 , N 1.01 ], and clearly we only need to consider 1/4 ≤ β ≤ 1.01. We count more solutions if we relax the condition to z m , z n ∈ [N β , 32N β ). Thus, let us consider (31) 1≤m,n≤M,m =n, for some β ∈ [1/4, 1.01]. We set up everything as in Case 1, but now we define T = 2 u N β . We define the b k 's as before but restricting ourselves to those z m contained in Note that previously we had k b k = M ≤ N 2 , whereas now we have a stronger bound. Applying the Cauchy-Schwarz inequality we obtain We define (a h ) h≥0 and P (t) as in Case 1 (see (15) and (16)). In the present case the inequality |j 1 z m −j 2 z n | < 1 is only possible when zm zn − j 2 j 1 ≤ 1 2 u−1 N β . By construction, 2 u−1 N β becomes large in comparison with T , and we can continue to argue as in Case 1. Note that now, as a consequence of (32), we have |P (0)| 2 ≪ E * N N β instead of |P (0)| 2 ≪ N 4 as in Case 1. Proceeding as in Case 1 we again obtain Recall that we only need to consider β ≤ 1.01, and that there are at most ≪ log N many different values of β to consider for Case 2. Inserting E * N ≪ N 220/91 and assuming that ε is small shows that the total contribution of Case 2 is of order ≪ N 3.93 , for every fixed value of u.

Proof of Theorem 1, Part 4: conclusion of the proof
The crucial ingredient in the proof of Theorem 1 is the variance bound coming from Section 4. From the previous section and ( 12) we deduce that for the variance defined in (11) we have , for a small (fixed) constant ε > 0. Everything else now follows from a standard procedure.
To be a bit more specific, convergence in (6) can be established using the estimate for the expectations in Section 3, and using the variance bound (33) together with Chebyshev's inequality and the Borel-Cantelli lemma. From that we get a convergence result for almost all α ∈ R, for fixed values of r and s. One notes that there are only countably many possible values of r, and that by continuity/monotonicity it is sufficient to consider countably many values of s. Since a countable union of sets of measure zero has measure zero as well, almost all α ∈ R have the property that (6) holds for all r and all s, as desired. We refer the reader to [5] or [34], where this argument is carried out in full detail. It applies without any modifications to the situation in the present paper.

Proof of Theorem 2.
Throughout this section we assume that θ > 1 is fixed, and we consider the sequence (x n ) defined by x n = n θ , n ≥ 1. Note that with this definition we have, for every n ≥ 1, 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 our proof of Theorem 2 we will rely on the following lemma of Robert and Sargos.
Then for every ε > 0, The restriction to a dyadic range for (n 1 , n 2 , n 3 , n 4 ) in the statement of the lemma does not actually play a role. This is easily seen by interpreting the number of solutions of the inequality as an L 4 -norm. Indeed, generalizing the definition in (7) and setting we have a measure whose Fourier transform is a (normalized) tent function on [−2γ, 2γ]. Let E * N,γ denote the number of solutions of (34), subject to (n 1 , n 2 , n 3 , n 4 ) ∈ {1, . . . , N } 4 . Assume 3 We thank Niclas Technau for pointing out to us that the estimate in Lemma 2 is also contained as a special case in a general result in a very recent paper of Huang [22]. Huang's result gives improved error terms, but for our application this does not play a role. However, the generality of Huang's results could allow further applications of our method in the spirit of our Theorem 2.
for simplicity of writing that N is a power of 2, i.e. N = 2 L for some L ≥ 1. Then applying Hölder's inequality we obtain which is obtained by interpreting the integrals in line (35) in terms of solutions of the Diophantine inequality (34), and applying Lemma 2 with parameters 2γ and M = 2 ℓ−1 . Thus we have (36) E * N,γ ≪ ε N 2+ε + γN 4−θ+ε for any ε > 0. Note that for θ ≥ 2 this bound is so small (for γ = 1) that we can apply Theorem 1 and derive the desired conclusion. Thus in the sequel we can restrict ourselves to the case θ < 2.
We set up the same machinery as in the proof of Theorem 1. Controlling the expectations, as in Section 3 above, is unproblematic. The crucial part is again the variance estimate. As in Section (4), we are led to the counting problem 1≤m,n≤M, m =n 2 u−1 ≤j 1 ,j 2 <2 u 1 |j 1 z m − j 2 z n | < 1 where now {z 1 , . . . , 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 ≤ 2rN , and M = N 2 − 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 } < 4N 1/4 . Thus again we can localize z m and z n , and restrict ourselves to counting (37) 1≤m,n≤M, m =n, where in the present situation we only need to consider values of β in the range from 1/4 to θ. Note that at most ≪ log N many different values of β need to be considered to cover the whole relevant range. Let β ∈ [1/4, θ] be fixed. Let u in (37) be fixed. For integers k ≥ 0 we define (38) Note the difference in comparison with (14). 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 (34) we can now control the number of solutions on a finer scale, and can accordingly set shorter ranges for the grouping of the z m 's (where γ = 2 −u ).
Let ε > 0 be a small constant, satisfying 3ε ≤ θ − 1. Set T = 2 u N min{β−ε,1+2ε} . Unlike the argument in the general case in Section 4, we do not explicitly distinguish between Case 1 and Case 2 ("large" and "small"), but have implicitly included this distinction into the way that T is defined. As in Section 4, we split the interval [1, ∞) into a disjoint union ∞ h=0 I h , where and set as well as Then by construction we again have as in Section 4.
From the particular structure and growth of the sequence (n θ ) we can easily see that so that now we have |P (0)| ≪ N 2+β−θ .
As in Section 4, we assume w.l.o.g. that j 1 ≥ j 2 . Recall that j 1 , j 2 ≥ 2 u−1 by assumption. Assume that z m ∈ [k/2 u , (k + 1)/2 u ) for some k. Then the inequality |j 1 z m − j 2 z n | ≤ 1 is only possible when which is a version of (18) that is adapted to the construction in (38). Arguing as in the lines leading to (19), this again gives which perfectly resembles (19) but where now the b k 's and a h 's are defined in a different way according to (38). Note that (36). Note also that T is chosen in such a way that j 1 z m and j 2 z n exceed T ; indeed, by assumption j 1 , j 2 ≥ 2 u−1 and z m , z n ≥ N β , while T ≤ 2 u N β−ε by definition. Thus we can continue the argument as in Section 5. It turns out that in this setting it is sufficient in order to bound Int 1 to use the crude bound |ζ(1/2 + it)| ≪ |t| 1/6 , rather than the more elaborate argument relying on estimates for moments of the Riemann zeta function. We obtain where we used that β ≤ θ ≤ 2, θ ≥ 1 + 3ε, T ≪ 2 u N 1+2ε , 2 u ≪ N . Noting that we need to 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 of Theorem 1. We remark that any bound better than the convexity bound for the Riemann zeta function would be sufficient to derive the same conclusion.

Closing remarks
As remarked in the introduction, our method breaks down completely when the growth order of the sequence (x n ) n≥1 is only linear or even slower. Not only does the "lattice point counting with the zeta function" argument from Section 5 fail to work in this situation, but there is a much more fundamental reason why the whole approach based on calculating first and second moments (expectations and variances, as in Sections 3 and 4) fails to work in this setup. To give a brief sketch of what causes the problem, assume that (x n ) n is a sequence of reals such that x n ≤ n, n ≥ 1. Assume that we want to bound the variance in analogy with (11), so say we want to show that tends to zero as N → ∞ (where we write the original indicator function instead of its approximation by a trigonometric polynomial, and where for simplicity of writing we set s = 1). By our assumption on the growth of (x n ) n , all differences x m − x n appearing in the sum above are uniformly bounded by N .
Thus the variance fails to tend to zero for a slowly growing (x n ) n , due to the fact the contribution of small values of α to the variance integral is too large. 4 The argument used in Section 5 fails to work in a similar way for slowly growing (x n ) n , since the error terms coming from the contribution to the integrals of values of t near zero become too large. 5 Consequently, it seems that 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 values of α near zero from the variance integral (41) by using a different measure which vanishes 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 with slowly growing sequences (x n ) n are a novel aspect which only shows up in the real-number setup -in contrast, when (a n ) n is an integer sequence which grows at most linearly, then (a n α) n fails to have Poissonian pair correlation for any α because the additive energy of (a 1 , . . . , a N ) necessarily is of maximal possible order (cf. [25]).
We formulate the following open problems.
Open Problem 1: Let θ ∈ (0, 1). Show that (n θ α) n≥1 has Poissonian pair correlation for almost all α. Note that Lemma 2 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 correlation for almost all α. We note that it is possible to establish a variant of Lemma 2 for this setting (with exponent 3 in place of 4 − θ). 4 A similar argument appears at the end of [32], where it is used to show that the L 2 approach fails to work in the case of the triple correlation of (n 2 α)n; cf. also [39] 5 It might be difficult to spot at a quick glance, so we briefly comment on where the speed of growth of (xn)n was used in our argument in Sections 5 and 7. There is a term |P (0)| 2 N 1+ε/4 coming from the contribution of values of t near the origin to the integral. This term is divided by T at the end of the calculation, so we cannot take T too small since we need N |P (0)| 2 N 1+ε/4 /N 4 T → 0. On the other hand, we cannot take T too large, since we need T ≪ 2 u zn to be able to detect the solutions of our Diophantine inequality. To balance everything out, we need to be able to assure that there are not too many small values of zn (i.e., not too many differences xm − xn which are "small"). In our proof of Theorem 1 our assumption on the order of the additive energy takes care of this: it is easy to see that an upper bound on E * N implies an upper bound on the number of "small" differences xm − xn; this is what we used in Case 2 of Section 5. In the setting of Theorem 2 we can control the number of small differences xm − xn efficiently because of the particularly simple structure of the sequence. The relevant equations are (32) and (39).
Open Problem 3: Let x n = n log n, n ≥ 1. Show that (x n α) n has Poissonian pair correlation for almost all α.
Clearly the exponent 220/91 − δ in the statement of Theorem 1 is not optimal. Most likely this can be improved to 3 − δ (which is the case conditionally under the Lindelöf hypothesis). It seems to us that the method of Bloom and Walker [8], which led to a quantitative improvement of the results of [5], cannot be used here. Their method relied on sum-product estimates, which, roughly speaking, leads to an integrand |P (t)| 2 being replaced by |P (t)| 4 . In the case of integer sequences (when working with the random model of the zeta function) one has perfect orthogonality, so that |P | 4 can be efficiently bounded. In our setting the situation is quite different -we have constructed our function P (t) in such a way that the diagonal contribution dominates when calculating |P | 2 , but we do not have orthogonality for |P | 4 and cannot efficiently bound this integral.
Open Problem 4: Show that Theorem 1 remains valid under the weaker assumption E * N ≪ N 3−δ for some δ > 0. Show that this can be further relaxed to assuming E * N,γ ≪ γN 4−δ , for all γ in a range from roughly 1/N to 1. It might even be the case that only values of γ near a 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 Poissonian pair correlation for almost all α when E N ≫ N 3 . It is unclear if there is a similar phenomenon in the real-number case.
Open Problem 5: Show that unlike in the integer case, it is possible for an increasing sequence (x n ) n≥1 of reals that E * N ≫ N 3 and that (x n α) n has Poissonian pair correlation for almost all α (compare Open Problems 1 and 2 above, where E * N ≫ N 3 ). Establish a criterion (stated for example in terms of E * N,γ ) which ensures that (x n α) n does not have Poissonian pair correlation for almost all α. A candidate for such a criterion is that E * N,γ ≫ γN 4 for some γ = γ(N ) for infinitely many N , where maybe one also has to assume that these values of γ are of size γ ≈ 1/N .