Poissonian Pair Correlations Research Articles

On exceptional sets in the metric Poissonian pair correlations problem

Let left( a_{n}right) _{n} be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy Eleft( A_{N}right) of the cut-offs A_{N}=left{ a_{n},{:},,nle Nright} , 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[ 0,1right] satisfying that left( leftlangle alpha a_{n}rightrangle right) _{n} with Eleft( 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.

Poissonian pair correlation and discrepancy

A sequence (xn)n=1∞ on the torus T≅[0,1] is said to exhibit Poissonian pair correlation if the local gaps behave like the spacings of a Poisson random variable, i.e. limN→∞1N#1≤m≠n≤N:|xm−xn|≤sN=2salmost surely.We show that being close to Poissonian pair correlation for few values of s is enough to deduce global regularity statements: if, for some 0<δ<1∕2, a set of points x1,…,xN satisfies 1N#1≤m≠n≤N:|xm−xn|≤sN≤(1+δ)2sfor all1≤s≤(8∕δ)logN,then the discrepancy DN of the set satisfies DN≲δ1∕3+N−1∕3δ−1∕2. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation N2DN5≲2N∫0N∕21N#1≤m≠n≤N:|xm−xn|≤sN−2s2ds≲N2DN2,where the lower bound is conditioned on DN≳N−1∕3. The proofs use a connection between exponential sums, the heat kernel on T and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of {〈αa(x)〉} x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that {〈αx d 〉} fails to have Poissonian pair correlation is at most $$\frac{{d + 2}}{{d + 3}} < 1$$ . This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $$\frac{2}{{d + 1}}$$ . An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved.

Pair correlations and equidistribution

A deterministic sequence of real numbers in the unit interval is called equidistributed if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs xk,xl∈(xn)1≤n≤N which are within distance s/N of each other is asymptotically ∼2sN. A randomly generated sequence has both of these properties, almost surely. There seems to be a vague sense that having Poissonian pair correlations is a “finer” property than being equidistributed. In this note we prove that this really is the case, in a precise mathematical sense: a sequence whose asymptotic distribution of pair correlations is Poissonian must necessarily be equidistributed. Furthermore, for sequences which are not equidistributed we prove that the square-integral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations.

On pair correlation and discrepancy

We say that a sequence left( x_nright) _{n ge 1} in [0, 1) has Poissonian pair correlations if limN→∞1N#1≤l≠m≤N:|xl-xm|<sN=2s\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\lim _{N \\rightarrow \\infty } \\frac{1}{N} \\# \\left\\{ 1 \\le l \\ne m \\le N{:}\\,\\left||x_l-x_m\\right|| < \\frac{s}{N} \\right\\} = 2s \\end{aligned}$$\\end{document}for all s>0. In this note we show that if the convergence in the above expression is—in a certain sense—fast, then this implies a small discrepancy for the sequence left( x_nright) _{n ge 1}. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).

Localized quantitative criteria for equidistribution

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length&nbsp;1). We show that if there exists a sequence $(t_n)_{n=1}^{\infty}$ of positive real numbers converging to 0 such that $$ \lim_{N \rightarrow \infty}{ \frac{1}{N^

On the pair correlation for fractional parts of vector sequences

We study the pair correlation for fractional parts of vector sequences and provide a sufficient condition for being poissonian. As a consequence, a large class of sequences of fractional parts of integer polynomials in several variables, lacunary integer sequences and linear combinations between them are shown to have poissonian pair correlation.