Poissonian Pair Correlations Research Articles

Poissonian pair correlation for higher dimensional real sequences

AbstractIn this article, we examine the Poissonian pair correlation (PPC) statistic for higher dimensional real sequences. Specifically, we demonstrate that for , almost all , the sequence in has PPC conditionally on the additive energy bound of . This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [Aistleitner, El‐Baz, and Munsch, Geom. Funct. Anal. 31 (2021), 483–512]. More generally, we derive the PPC for for almost all . As a consequence we establish the metric PPC for provided that all of the are greater than two.

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

Abstract We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$ -horospheres in the space of affine lattices.

The pair correlation function of multi-dimensional low-discrepancy sequences with small stochastic error terms

In any dimension d≥2, there is no known example of a low-discrepancy sequence which possesses Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have β-Poissonian pair correlations for all 0<β<1d and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case β=1d). In this paper, we further elaborate on the closeness of the two notions. We show that d-dimensional Kronecker sequences for badly approximable vectors α→ with an arbitrary small uniformly distributed stochastic error term generically have β=1d-Poissonian pair correlations.

Poissonian Pair Correlation for $\alpha n^{\theta }$ mod 1

Abstract We show that sequences of the form $\alpha n^{\theta } \pmod {1}$ with $\alpha&amp;gt; 0$ and $0 &amp;lt; \theta &amp;lt; \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $ have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for $\alpha&amp;gt; 0$ and $0 &amp;lt; \theta &amp;lt; \tfrac {14}{41} = \tfrac {1}{3} + 0.0081 \ldots $. We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri–Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert–Sargos and (Fouvry–Iwaniec–)Cao–Zhai. The exponent $\theta = \tfrac {2}{5}$ is the limit of our approach.

On higher dimensional Poissonian pair correlation

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any d≥2, strictly increasing sequences (an(1)),…,(an(d)) of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is O(N3−δ) for any δ>0. Further, in dimension two, we establish an analogous result with respect to the 2-norm. As a consequence, it follows that ({nα},{n2β}) and ({nα},{[nlogA⁡n]β}) (A∈[1,2]) have Poissonian pair correlation for almost all (α,β)∈R2 with respect to sup-norm and 2-norm. This gives a negative answer to the question raised by Hofer and Kaltenböck [15]. The proof uses estimates for ‘Generalized’ GCD-sums.

Triple correlation and long gaps in the spectrum of flat tori

We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequences we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most (resp. at least) Poissonian, and that (b) almost all flat tori contain infinitely many gaps in their spectrum that are at least 2.006 times longer than the average gap. The significance of the constant 2.006 lies in the fact that there exist sequences with Poissonian pair correlation and with consecutive spacings bounded uniformly from above by 2, as we also prove in this paper. Thus our result goes beyond what can be deduced solely from the Poissonian behavior of the pair correlation.

Some connections between discrepancy, finite gap properties, and pair correlations

A generic uniformly distributed sequence (x_n)_{n in mathbb {N}} in [0, 1) possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of alpha -pair correlations with 0< alpha < 1. Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least alpha -pair correlations for 0< alpha <1. According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have alpha -pair correlations for 0< alpha < 1.

Sequences with almost Poissonian pair correlations

Although a generic uniformly distributed sequence has Poissonian pair correlations, only one explicit example has been found up to now. Additionally, it is even known that many classes of uniformly distributed sequences, like van der Corput sequences, Kronecker sequences and LS sequences, do not have Poissonian pair correlations. In this paper, we show that van der Corput sequences and the Kronecker sequence for the golden mean are as close to having Poissonian pair correlations as possible: they both have α-pair correlations for all 0<α<1 but not for α=1 which corresponds to Poissonian pair correlations.

On the number of gaps of sequences with Poissonian pair correlations

A sequence (xn) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖xi−xj‖≤s/N}=2sN(1+o(1)) for all reals s>0, as N→∞. It is known that, if (xn) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x1,…,xn} cannot be bounded along any index subsequence (nt). First, we improve this by showing that, if (xn) has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of {x1,…,xn} is o(n), as n→∞. Furthermore, we show that for every function f:N+→N+ with limn⁡f(n)=∞ there exists a sequence (xn) with Poissonian pair correlations such that g(n)≤f(n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.

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 ge 1} has been pioneered by Rudnick, Sarnak and Zaharescu. Here alpha is a real parameter, and (a_n)_{n ge 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 ge 1}. In the present paper we develop a similar framework for the case when (a_n)_{n ge 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 ge 1} has Poissonian pair correlation for almost all alpha in {mathbb {R}}.

A note on sequences not having metric Poissonian pair correlations

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed that all sequences have MPPC. A similar result appears in work of Lachmann and Technau and is proved using a totally different strategy. The main novelty here is the simplicity of the proof, which we arrive at by modifying a construction of Bourgain.

Pair correlations of Halton and Niederreiter Sequences are not Poissonian

Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences—even though they are uniformly distributed—fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

Poissonian correlation of higher order differences

A sequence (xn)n=1∞ on the torus T exhibits Poissonian pair correlation if for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this condition implies equidistribution of (xn). We generalize this result to four-fold differences: if for all s>0 we havelimN→∞⁡1N2#{1≤m,n,k,l≤N{m,n}≠{k,l}:|xm+xn−xk−xl|≤sN2}=2s then (xn)n=1∞ is equidistributed. This notion generalizes to higher orders, and for any k we show that a sequence exhibiting 2k-fold Poissonian correlation is equidistributed. In the course of this investigation we obtain a discrepancy bound for a sequence in terms of its closeness to 2k-fold Poissonian correlation. This result refines earlier bounds of Grepstad & Larcher and Steinerberger in the case of pair correlation, and resolves an open question of Steinerberger.

Equidistribution results for sequences of polynomials

Let (fn)n=1∞ be a sequence of polynomials and α>1. In this paper we study the distribution of the sequence (fn(α))n=1∞ modulo one. We give sufficient conditions for a sequence (fn)n=1∞ to ensure that for Lebesgue almost every α>1 the sequence (fn(α))n=1∞ has Poissonian pair correlations. In particular, this result implies that for Lebesgue almost every α>1, for any k≥2 the sequence (αnk)n=1∞ has Poissonian pair correlations.

Some negative results related to Poissonian pair correlation problems

We say that a sequence (xn)n∈N in [0,1) has Poissonian pair correlations if limN→∞1N#1≤l≠m≤N:‖xl−xm‖≤sN=2sfor every s≥0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn)n∈N of real numbers in [0,1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and more generally LS-sequences of points and digital (t,1)-sequences. Additionally, this theorem enables us to derive negative pair correlation results for sequences of the form ({anα})n∈N, where (an)n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α.Second, in this note we study the pair correlation statistic for sequences of the form, xn={bnα},n=1,2,3,…, with an integer b≥2, where we choose α as the Stoneham number or as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property.

Poissonian pair correlation in higher dimensions

Let (xn)n=1∞ be a sequence on the torus T (normalized to length 1). A sequence (xn) is said to have Poissonian pair correlation if, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this implies uniform distribution of the sequence (xn). Hinrichs, Kaltenböck, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences (xn) in [0,1]d that satisfy, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖∞≤sN1/d}=(2s)d are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence (xn) in Td satisfies, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖2≤sN1/d}=ωdsd, where ωd is the volume of the unit ball, then (xn) is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

Low discrepancy sequences failing Poissonian pair correlations

M. Levin defined a real number x that satisfies that the sequence of the fractional parts of $$(2^n x)_{n\ge 1}$$ are such that the first N terms have discrepancy $$O((\log N)^2/ N)$$ , which is the smallest discrepancy known for this kind of parametric sequences. In this work we show that the fractional parts of the sequence $$(2^n x)_{n\ge 1}$$ fail to have Poissonian pair correlations. Moreover, we show that all the real numbers x that are variants of Levin’s number using Pascal triangle matrices are such that the fractional parts of the sequence $$(2^n x)_{n\ge 1}$$ fail to have Poissonian pair correlations.

The Champernowne constant is not Poissonian

We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \rbrace = 2s \end{equation*} for every $s \geq 0$. In this note we study the pair correlation statistics for the sequence of shifts of $\alpha$, $x_n = \lbrace 2^n \alpha \rbrace, \ n=1, 2, 3, \ldots$, where we choose $\alpha$ as the Champernowne constant in base $2$. Throughout this article $\lbrace \cdot \rbrace$ denotes the fractional part of a real number. It is well known that $(x_n)_{n \in \mathbb{N}}$ has Poissonian pair correlations for almost all normal numbers $\alpha$ (in the sense of Lebesgue), but we will show that it does not have this property for all normal numbers $\alpha$, as it fails to be Poissonian for the Champernowne constant.

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1 / N. In the d-dimensional case, of course, the order of the mean spacing is 1/N^{frac{1}{d}}, and—in our concept—the distance of sequence elements will be measured by the supremum-norm. Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

Pair correlation of sequences with maximal additive energy

AbstractWe show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.