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Effective pair correlations of fractional powers of integers

Rafael Sayous

TL;DR

This work analyzes pair correlations of the deterministic sequence $u_n=n^\alpha$ for $\alpha\in(0,1)$ by introducing empirical measures ${\cal R}_N^{\nalpha}$ with a scaling factor $\phi(N)$ and renormalization $\psi(N)$. It proves vague convergence ${\cal R}_N^{\nalpha} \overset{*}{\rightharpoonup} \rho_\nalpha \;\mathrm{Leb}_{\mathbb{R}}$ where the limiting density $\rho_\nalpha$ depends on $\lambda=\lim_{N\to\infty}\frac{\phi(N)}{N^{1-\nalpha}}$, showing mass loss when $\lambda=+\infty$, Poissonian statistics when $\lambda=0$, and an exotic level-repulsion density for $0<\lambda<\infty$. In the critical scaling $\phi(N)\sim N^{1-\nalpha}$, the density decays with a power-law tail and features a pronounced level repulsion at the origin, with the exceptional case $\lambda=1$ interpreted via an unfolding approach. The paper provides explicit error terms and an unfolding construction that relates the exotic density to a higher-dimensional limit, yielding a comprehensive picture of how deterministic sequences transition from Poisson-like to repulsive correlation structures under different scalings.

Abstract

We study the statistics of pairs from the sequence $(n^α)_{n\in\mathbb{N}^*}$, for every parameter $α\in \, ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor $N\mapsto N^{1-α}$, we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence.

Effective pair correlations of fractional powers of integers

TL;DR

This work analyzes pair correlations of the deterministic sequence for by introducing empirical measures with a scaling factor and renormalization . It proves vague convergence where the limiting density depends on , showing mass loss when , Poissonian statistics when , and an exotic level-repulsion density for . In the critical scaling , the density decays with a power-law tail and features a pronounced level repulsion at the origin, with the exceptional case interpreted via an unfolding approach. The paper provides explicit error terms and an unfolding construction that relates the exotic density to a higher-dimensional limit, yielding a comprehensive picture of how deterministic sequences transition from Poisson-like to repulsive correlation structures under different scalings.

Abstract

We study the statistics of pairs from the sequence , for every parameter . We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor , we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence.
Paper Structure (10 sections, 6 theorems, 87 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main statement and technical lemmas
  3. Symmetry of the empirical pair correlation measures
  4. Linear approximation
  5. Riemann sum approximation for compactly supported functions
  6. Proof of Theorem \ref{['th:effective_cv_correlations']}
  7. Regime $\frac{\phi(N)}{N^{1-\alpha}} \underset{N \to \infty}{\longrightarrow}+\infty$
  8. Regime $\frac{\phi(N)}{N^{1-\alpha}} \underset{N\to\infty}{\longrightarrow}0$
  9. Regime $\frac{\phi(N)}{N^{1-\alpha}} \underset{N\to\infty}{\longrightarrow}\lambda \in {\mathbb R}_+^*$
  10. Unfolding technique

Key Result

Theorem 1.1

We have the following vague convergence of positive measures where $\operatorname{Leb}_{\mathbb R}$ is the Lebesgue measure on ${\mathbb R}$ and $\rho_\alpha : {\mathbb R} \to {\mathbb R}_+$ is the measurable nonnegative function given by where $|\cdot|$ denotes the absolute value function on ${\mathbb R}$, and $\lfloor*\rfloor{\cdot}$ is the lower integer part function from ${\mathbb R}$ to ${\

Figures (2)

  • Figure 1: The empirical distribution (in blue) of pair correlations for $(\sqrt{n})_{1\leq n\leq N}$ with $N=10^6$ using the scaling factor $N\mapsto \sqrt{N}$ (and renormalization factor $N\mapsto N$), and the limit distribution $\rho_{\frac{1}{2}}$ (in red).
  • Figure 2: The scaled pair correlation functions $\widetilde{\rho_{\alpha}}$ in the exotic case $\lambda=1$ for different power parameters: $\alpha = \frac{1}{2}$ (in blue), $\alpha=\frac{9}{10}$ (in green) and $\alpha=\frac{99}{100}$ (in red).

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Remark 2.6
  • proof
  • Lemma 2.7
  • ...and 1 more