# On exceptional sets in the metric Poissonian pair correlations problem

## Abstract

Let \(\left( a_{n}\right) _{n}\) be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy \(E\left( A_{N}\right) \) of the cut-offs \(A_{N}=\left\{ a_{n}\,{:}\,\,n\le N\right\} \), and \(\left( a_{n}\right) _{n}\) possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of “second order”. Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of \(\alpha \in \left[ 0,1\right] \) satisfying that \(\left( \left\langle \alpha a_{n}\right\rangle \right) _{n}\) with \(E\left( A_{N}\right) =\Omega \left( N^{3}\right) \) does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.

## Keywords

Poissonian pair correlations Additive energy Diophantine approximation Metric number theory## Mathematics Subject Classification

Primary 11B30 Secondary 11B25 11K06 11K60## 1 Introduction

The theory of uniform distribution modulo 1 dates back, at least, to the seminal paper [23] of Weyl who showed that for any fixed \(\alpha \in \mathbb {R}{\setminus }\mathbb {Q}\) and integer \(d\ge 1\) the sequences \(\left( \bigl \langle \alpha n^{d}\bigr \rangle \right) _{n}\) are uniformly distributed modulo 1 where \(\left\langle x\right\rangle \) denotes the fractional part of \(x\in \mathbb {R}\). However, in recent years various authors [2, 4, 5, 9, 11, 14, 15, 16, 17, 18, 21, 22] have been investigating a more subtle distribution property of such sequences—namely, whether the asymptotic distribution of the pair correlations has a property which is called Poissonian, and defined as follows.

### Definition

^{1}

*s*as \(N\rightarrow \infty \). Moreover, let \(\left( a_{n}\right) _{n}\) denote a strictly increasing sequence of positive integers. If no confusion can arise, we write

*not*have PPC for

*any*\(\alpha \in \mathbb {R}\) if \(d=1\). For \(d\ge 2\), Rudnick and Sarnak [16] proved that \(\left( n^{d}\right) _{n}\) has metric Poissonian pair correlations (metric PPC). A result of Aistleitner et al. [2], who used a Fourier analytic approach combined with a bound on GCD sums of Bondarenko and Seip [6], uncovered the connection of the metric PPC property of \(\left( a_{n}\right) _{n}\) with its combinatoric properties. For stating it, we introduce some notation. Let \(\left( a_{n}\right) _{n}\) denote throughout this article a strictly increasing sequence of positive integers, and abbreviate the set of the first

*N*elements of \(\left( a_{n}\right) _{n}\) by \(A_{N}\). Moreover, define the additive energy \(E\left( I\right) \) of a finite set of integers

*I*via

*S*. In the following, let \(\mathcal {O}\) and

*o*denote the Landau symbols/O-notation, and \(\ll \) or \(\gg \) the Vinogradov symbols. The dependence of an implied constant in one of these symbols will be indicated by mentioning this parameter in a subscript.

*I*has large additive energy if and only if it contains a “large” arithmetic progression like structure. Indeed, if \(\left( a_{n}\right) _{n}\) is a geometric progression or of the form \(\left( n^{d}\right) _{n}\) for \(d\ge 2,\) then (2) is satisfied.

Recently, Bloom, Chow, Gafni, Walker relaxed—provided that, roughly speaking, the density of the sequence does not decay faster than \(1/(\log N)^{2}\)—the power saving bound (2) for detecting the metric PPC property of \(\left( a_{n}\right) _{n}\) significantly:

### Theorem A

In accordance with probabilistic considerations, cf. [4, Thm. 1.5], the above result could be seen as a sign of Khintchine-type law underpinning the characterization of the metric PPC property of \(\left( a_{n}\right) _{n}\). Indeed, the following basic question about the nature of the connection between additive energy and the metric PPC property was raised in [4]:

**Fundamental Question** [4] Is it true that if \(E\left( A_{N}\right) \sim N^{3}\psi \left( N\right) \) for some weakly decreasing function \(\psi \,{:}\,\mathbb {N}\rightarrow \left[ 0,1\right] \), then \(\left( a_{n}\right) _{n}\) has metric PPC if and only if \(\sum _{N\ge 1}\psi \left( N\right) /N\) converges?

### Remark

This question will be answered in the negative in a forthcoming note of Aistleitner and the authors.

Regarding the optimal bound for \(E\left( A_{N}\right) \) to ensure the metric PPC property of \(\left( a_{n}\right) _{n}\), the following two questions were raised in [2]. For stating those, we use the convention that \(f=\Omega \left( g\right) \) means for \(f,g\,{:}\,\mathbb {N}\rightarrow \mathbb {R}\) there is a constant \(c>0\) such that \(g\left( n\right) >cf\left( n\right) \) holds for infinitely many *n*.

### Question 1

Is it possible for \(\left( a_{n}\right) _{n}\) with \(E\left( A_{N}\right) =\Omega \left( N^{3}\right) \) to have metric PPC?

### Question 2

Do all \(\left( a_{n}\right) _{n}\) with \(E\left( A_{N}\right) =o\left( N^{3}\right) \) have metric PPC?

^{2}whether the sequence of prime numbers \(\left( p_{n}\right) _{n}\), ordered by increasing value, has metric PPC. Recently, Walker [22] answered this question in the negative by showing that there is a constant \(c>0\) satisfying that for almost every \(\alpha \in \left[ 0,1\right] \) the inequality \(R\left( \left[ -s,s\right] ,\alpha ,N\right) >c\) holds for infinitely many

*N*. Thereby he gave a significantly better bound than (3) for the additive energy \(E\left( A_{n}\right) \) for a sequence \(\left( a_{n}\right) _{n}\) not having metric PPC—since the additive energy of the truncations of \(\left( p_{n}\right) _{n}\) is in \(\Theta \bigl (\left( \log N\right) ^{-1}N^{3}\bigr )\) where \(f=\Theta \left( g\right) \), for functions

*f*,

*g*, means that both \(f=\mathcal {O}\left( g\right) \) and \(g=\mathcal {O}\left( f\right) \) holds.

For a given sequence \(\left( a_{n}\right) _{n}\), we denote by \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) \) the “exceptional” set of all \(\alpha \in \left( 0,1\right) \) such that \(\left( \left\langle \alpha a_{n}\right\rangle \right) _{n}\) does not have PPC.

### Theorem B

[2] If \(E(A_{N})=\Omega \left( N^{3}\right) \), then \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) \) has positive Lebesgue measure.

We prove the following sharpening.

### Theorem 1

If \(E(A_{N})=\Omega \left( N^{3}\right) \), then \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) \) has full Lebesgue measure.

Some remarks are in order.

### Remark

- (a)
\(\left( a_{n}\right) _{n}\) is called quasi-arithmetic of degree one, cf. [1, Def. 1], if infinitely often at least a constant proportion of elements of \(A_{N}\) is contained in some arithmetic progression of length \(\ll N\). Any such sequence obviously satisfies \(E\left( A_{N}\right) =\Omega \left( N^{3}\right) \). Theorem 1 improves upon a recent result of Aichinger, Aistleitner, and Larcher [1, Thm. 3] who showed that \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) \) has full Lebesgue measure, if \(\left( a_{n}\right) _{n}\) is quasi-arithmetic of degree one.

- (b)
Recently, Larcher [12, Thm. 1] sharpened this result to \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) =\left( 0,1\right) \), and subsequently Larcher and Stockinger [10, Thm. 1] extended this to quasi-arithmetic sequences of any degree \(d\ge 1\)—which due to Freiman’s theorem (cf. [12, text above Def. 2]) implies that if \(E(A_{N})=\Omega \left( N^{3}\right) \), then \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) =\left( 0,1\right) \).

For stating our second main theorem, we denote by \(\mathbb {R}_{>x}\) the set of real numbers exceeding a given \(x\in \mathbb {R}\).

### Theorem 2

*f*.

*r*-folded iterated logarithm is denoted by \(\log _{r}\left( x\right) \), i.e.

### Corollary 1

*r*be a positive integer. Then, there is a strictly increasing sequence \(\left( a_{n}\right) _{n}\) of positive integers with

The proof of Theorem 2 connects the metric PPC property to the notion of optimal regular systems from Diophantine approximation. It uses, among other things, a Khintchine-type theorem due to Beresnevich. Furthermore, despite leading to better bounds, the nature of the sequences underpinning Theorem 2 is much simpler than the nature of those sequences previously constructed by Bourgain [2] (who used, inter alia, large deviations inequalities from probability theory), or the sequence of prime numbers studied by Walker [22] (who relied on estimates, derived by the circle-method, on the exceptional set in Goldbach-like problems).

## 2 First main theorem

Let us give an outline of the proof of Theorem 1. For doing so, we begin by sketching the reasoning of Theorem B: As it turns out, except for a set of negligible measure, the counting function in (1) can be written as a function (of \(\alpha \)) that admits a non-trivial estimate for its \(L^{1}\)-mean value. The mean value is infinitely often too small on sets whose measure is uniformly bounded from below. Thus, there exists a sequence of sets \(\left( \Omega _{r}\right) _{r}\) of \(\alpha \in \left[ 0,1\right] \) such that \(R\left( \left[ -s,s\right] ,\alpha ,N\right) \) is too small for every \(\alpha \in \Omega _{r}\) for having PPC and Theorem B follows.

*t*the relation

### 2.1 Preliminaries

We start with a well-know result relating, in a quantitative manner, the additive energy of a set of integers with the existence of a (relatively) dense subset with small difference set where the difference set \(B-B:=\left\{ b-b':\,b,b'\in B\right\} \) for a set \(B\subseteq \mathbb {R}\).

### Lemma 1

*c*such that the following holds. If \(E(A)\ge c\left( \#A\right) ^{3}\), then there is a subset \(B\subseteq A\) such that

- 1.
\(\#B\ge c_{1}\#A,\)

- 2.
\(\#\left( B-B\right) \le c_{2}\#A.\)

### Lemma 2

### Proof

### Lemma 3

Suppose *A* is a finite intersection of Bohr sets, and *B* is a finite union of Bohr sets. Then, \(A{\setminus } B\) is the union of finitely many intervals.

Furthermore, we shall use the Borel–Cantelli lemma in a version due to Erdős, and Rényi.

### Lemma 4

*j*denotes a positive integer. In the first part of the argument, we show how a sequence—that is constructed in the second part of the argument—can be used to deduce Theorem 1. For every fixed

*j*, we find a corresponding \(s=s(j)\) and construct inductively a sequence \(\left( \Omega _{r}\right) _{r}\) of exceptional values \(\alpha \) with the following properties:

- (i)For all \(\alpha \in \Omega _{r}\), the pair correlation function admits the upper boundfor some absolute constant \(\tilde{c}\in \left( 0,1\right) \), depending on \(\left( a_{n}\right) _{n}\) only.$$\begin{aligned} R\left( \left[ -s,s\right] ,\alpha ,N\right) \le 2\tilde{c}s \end{aligned}$$(6)
- (ii)For all integers \(r>t\ge 1\), the relationholds.$$\begin{aligned} \lambda \left( \Omega _{r}\cap \Omega _{t}\right) \le \lambda \left( \Omega _{r}\right) \lambda \left( \Omega _{t}\right) +2\varepsilon \lambda \left( \Omega _{t}\right) +\mathcal {O}\left( r^{-2}\right) \end{aligned}$$(7)
- (iii)
Each \(\Omega _{r}\) is the union of finitely many intervals (hence measurable).

- (iv)For all \(r\ge 1\), the measure \(\lambda \left( \Omega _{r}\right) \) is uniformly bounded from below by$$\begin{aligned} \lambda \left( \Omega _{r}\right) \ge \frac{c_{1}^{2}}{8}. \end{aligned}$$(8)

### 2.2 Proof of Theorem 1

*N*. By choosing an appropriate subsequence \(\left( N_{i}\right) _{i}\) and omitting the subscript

*i*for ease of notation, we may suppose that \(E\left( A_{N}\right) >cN^{3}\) holds for every

*N*occurring in this proof. Moreover, let \(c_{1},c_{2}\) and \(B_{N}\) be as in Lemma 1, corresponding to the

*c*just mentioned. Let

^{3}we assume that there are sets \(\left( \Omega _{r}\right) _{1\le r<R}\) given that satisfy the properties (i)–(iv) for all distinct integers \(1\le r,t<R\). Let \(N\ge R\). Since, due to Lemma 1,

*x*), i.e.

*xN*choices of \(\left( r,t\right) \in \mathcal {D}_{N}\) such that \(\left| r-t\right| \le x\), we obtain

*N*is sufficiently large. Moreover, for any \(\left| r-t\right| >PR^{k}\) we observe that

*k*, the lower bound

*I*be a subinterval of \(\Omega _{r}\). Then,

*N*denote the smallest integer

*m*with \(E\left( A_{m}\right) >cm^{3}\). We replace \(\mathcal {P}_{i}\) in (9) by \(\left[ 0,1\right] \) to directly derive

Thus, the proof is complete.

## 3 Second main theorem

The sequences \(\left( a_{n}\right) _{n}\) enunciated in Theorem 2 are constructed in two steps. In the first step, we concatenate (finite) blocks, with suitable lengths, of arithmetic progressions to form a set \(P_{A}\). In the second step, we concatenate (finite) blocks, with suitable lengths, of geometric progressions to form a set \(P_{G}\) and then define \(a_{n}\) to be the *n*-th smallest element of \(P_{A}\cup P_{G}\). On the one hand, the arithmetic progression like part \(P_{A}\) serves to ensure, due to considerations from metric Diophantine approximation, the divergence property (5) on a set with full measure or controllable Hausdorff dimension; on the other hand, the geometric progression like part \(P_{G}\) lowers the additive energy, as much as it can. For doing so, a geometric block will appear exactly before and after an arithmetic block, and have much more elements.

*f*is as in Theorem 2. We set \(P_{A}^{\left( 1\right) }\) to be the empty set while \(P_{G}^{\left( 1\right) }:=\left\{ 1,2\right\} \). Suppose \(P_{A}^{\left( j-1\right) },P_{G}^{\left( j-1\right) }\)for \(j\ge 2\) are already constructed. Let \(C_{j}=2\max \bigl \{ P_{G}^{(j-1)}\bigr \}\). Then

^{4}to be chosen later-on. Letting

*n*-th smallest element in \(P_{A}\cup P_{G}\). For \(d\in \mathbb {Z}\) and finite sets of integers

*X*,

*Y*, we abbreviate the number of representations of

*d*as a difference of an \(x\in X\) and a \(y\in Y\) by

*X*and the pair correlation counting function can be written as

### 3.1 Preliminaries

For determining the order of magnitude of \(E\left( A_{N}\right) \), the following considerations are useful. Since the cardinality \(P_{G}^{\left( j\right) }\cup P_{A}^{\left( j\right) }\) has about exponential growth, it is reasonable to expect \(E\left( A_{N}\right) \) to be of the same order of magnitude as the additive energy of the last block \(P_{G}^{\left( J\right) }\cup P_{A}^{\left( J\right) }\) that is fully contained in \(A_{N}\)—note that \(J=J\left( N\right) \); i.e. to expect the magnitude of \(E\bigl (P_{G}^{\left( J\right) }\cup P_{A}^{\left( J\right) }\bigr )\) which is roughly \(E\bigl (P_{A}^{\left( J\right) }\bigr )\). The next proposition verifies this heuristic.

### Proposition 1

Let \(\left( a_{n}\right) _{n}\) be as in the beginning of Sect. 3, and *f* be as in one of the two assertions in Theorem 2. Then, \(E\left( A_{N}\right) =\Theta \bigl (N^{3}\bigl (f\bigl (N\bigr )\bigr ){}^{-3\left( \beta -\gamma \right) }\bigr )\).

For the proof of Proposition 1, we need the following technical lemma.

### Lemma 5

*f*is as in Proposition 1. Then, \(\sum _{i\le j}F_{i}=\mathcal {O}\bigl (F_{j}\bigr )\) and

### Proof

*f*is such that (4) converges and \(f\left( 2x\right) \le \left( 2-\varepsilon \right) f\left( x\right) \) for

*x*large enough, then we obtain by a similar argument that \(\sum _{i\le j}F_{i}\) is in \(\mathcal {O}\bigl (F_{j}\bigr )\). Furthermore, \(\mathrm {rep}_{P_{G}^{\left( j\right) },P_{A}^{\left( i\right) }}\left( d\right) =\mathcal {O}\left( i\right) \), for every \(j\ge 1\), and non-vanishing for \(\mathcal {O}\bigl (2^{2j}\bigr )\) values of

*d*which implies the last claim. \(\square \)

We can now prove the proposition.

### Proof of Proposition 1

*j*such that \(P_{G}^{\left( j-1\right) }\subseteq A_{N}\). By exploiting (12),

*d*as \(i,j\le J\). Since \(\mathrm {rep}_{P_{A}^{\left( i\right) },P_{A}^{\left( j\right) }}\left( d\right) \le F_{\min \left( i,j\right) }\) holds, we deduce that

*d*, we obtain that

For estimating the measure or the Hausdorff dimension of \(\mathrm {NPPC}\left( \left( a_{n}\right) _{n}\right) \) from below, we recall the notion of an optimal regular system. This notion, roughly speaking, describes sequences of real numbers that are exceptionally well distributed in any subinterval, in a uniform sense, of a fixed interval.

### Definition

*J*be a bounded real interval, and \(S=\left( \alpha _{i}\right) _{i}\) a sequence of distinct real numbers.

*S*is called an optimal regular system in

*J*if there exist constants \(c_{1},\,c_{2},\,c_{3}>0\)—depending on

*S*and

*J*only—such that for any interval \(I\subseteq J\) there is an index \(Q_{0}=Q_{0}\left( S,I\right) \) such that for any \(Q\ge Q_{0}\) there are indices

Moreover, we need the following result(s) due to Beresnevich which may be thought of as a far reaching generalization of the classical Khintchine theorem, and the Jarník-Besicovitch theorem in Diophantine approximation.

### Theorem 3

*i*. If

Conversely, if (17) converges, then \(\mathcal {K}_{S}\left( \psi \right) \) has measure zero and the Hausdorff dimension equals the reciprocal of the lower order of \(\frac{1}{\psi }\) at infinity.

For a rational \(\alpha =\frac{p}{q}\), where \(p,q\in \mathbb {Z}, q\ne 0\), we denote by \(H\left( \alpha \right) \) its (naive) height, i.e. \(H\left( \alpha \right) :=\max \left\{ \left| p\right| ,\left| q\right| \right\} \). It is well-known that the set of rational numbers in \(\left( 0,1\right) \)—first running through all rationals of height 1 ordered by increasing numerical value, then through all rationals with height 2 ordered by increasing numerical value, and so on—gives rise to an optimal regular system in \(\left( 0,1\right) \). The following lemma says, roughly speaking, that this assertion remains true for the set of rationals in \(\left( 0,1\right) \) whose denominators are members of a special sequence that is not too sparse in the natural numbers, and hand-tailored for our purposes. The proof can be given by modifying the proof of the classical case, compare [7, Prop. 5.3]; however, we shall give the details for making this article more self-contained.

### Lemma 6

*S*is an optimal regular system in \(\left( 0,1\right) \).

### Proof

*X*. We take \(J=J\left( X\right) \) to be the largest integer \(j\ge 1\) such that \(B_{j}\le X\). Then, for

*X*large enough, there are at least, due to a basic property of the Eulerian totient function,

*X*. Hence, we obtain

*i*sufficiently large. Let \(Q\in \mathbb {N}, I\subseteq \left[ 0,1\right] \) be a non-empty interval, and let

*F*denote the set of \(\xi \in I\) satisfying the inequality \(\left\| q\xi \right\| <Q^{-1}\) with some \(1\le q\le \frac{1}{1000}Q\). Note that

*F*has measure at most

*J*as above with \(X=Q\) sufficiently large, it follows that

*S*is shown to be an optimal regular system. \(\square \)

Now we can proceed to the proof of Theorem 2.

### 3.2 Proof of Theorem 2

We argue in two steps depending on whether or not the series (4) converges. Proposition (1) implies the announced \(\Theta \)-bounds on the additive energy of \(A_{N}\), in both cases.

*J*, then, by the properties of \(\vartheta \) from Lemma 6 and the relation \(\sum _{j\le J}F_{j}=\mathcal {O}\left( F_{J}\right) \) from Lemma 5, we conclude that

*f*and \(\vartheta \left( x\right) =\mathcal {O}\bigl (x^{\nicefrac {1}{4}}\bigr )\) yields that if

*j*is large enough, then \(b_{j}\le n\le B_{j}\) implies \(cn^{2}>2^{j}\) and hence we obtain \(\psi \left( i\right) \le c^{-1}n^{-2}\bigl (f\bigl (2^{j}\bigr )\vartheta \bigl (2^{j}\bigr )\bigr )^{-1}\). Combining these considerations, we infer that

*i*and \(j=j\left( i\right) \). Now if \(b_{j}\le n\le B_{j}\) for

*j*sufficiently large and \(n,\alpha \) as in (19), then it follows that by taking any integer \(m\le \left( f\left( 2^{j}\right) \right) ^{\gamma }\bigl (\vartheta \bigl (2^{j}\bigr )\bigr )^{\nicefrac {1}{3}}\) that also the multiples

*j*sufficiently large. By (13), we obtain

*j*where \(C>0\) is some constant. For the optimal choice of the parameters \(\beta ,\gamma >0\), we are therefore led to maximize \(\beta -\gamma \) where \(2\gamma -\beta \ge 0\) and \(\gamma -1\le -\beta \) have to be satisfied. The solution is given if equality in the first inequality occurs, leading to \(\beta =\nicefrac {2}{3}\) and \(\gamma =\nicefrac {1}{3}\). Hence, (5) follows for \(\alpha \in \mathcal {K}_{S}\left( \psi \right) \).

## 4 Concluding remarks

It should be possible to relax the growth restriction \(f\left( x\right) =\mathcal {O}\bigl (x^{\nicefrac {1}{3}}\left( \log \left( x\right) \right) ^{-\nicefrac {7}{3}}\bigr )\) in Theorem 2 on the expense of some additional technical work; as the main objective in this section was to get as close as possible to the Khintchine-type threshold for making progress on the Fundamental Question, we have not expended much effort in possible relaxations.

We would like to mention an open problem related to this article. It asks about how much the PPC property is violated for a sequence that has not metric PPC.

### Problem

## Footnotes

- 1.
The subscript 2 in \(R_{2}\) indicates that relations of second order, i.e. pair correlations, are counted.

- 2.
This problem was posed at the problem session of the ELAZ conference in 2016.

- 3.
The base step uses simplified versions of the arguments exploited in the induction step, and will therefore be postponed.

- 4.
No particular importance should be attached to requiring \(\beta <\nicefrac {3}{4}\), or using “dyadic steps lengths \(2^{j}\)”. Doing so is for simplifying the technical details only—eventually, it will turn out that \(\beta =\nicefrac {2}{3}=2\gamma \) is the optimal choice of parameters in this approach. For proving this to the reader, we leave \(\gamma ,\beta \) undetermined till the end of this section.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). Both authors would like to express their gratitude towards C. Aistleitner for introducing them to the topic of this article, and valuable discussions. Thanks are also due to Thomas Bloom, Sam Chow, Ayla Gafni, and Aled Walker for helpful conversations.

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