2016/12/16 by Christoph Aistleitner, Aistleitner, Christoph, Thomas Lachmann +3 · 5 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1612.05495
openalex publication_date 2016/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.