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On the smallest gap in a sequence with Poisson pair correlations

Daniel Altman, Zachary Chase

TL;DR

This work addresses how small the largest gap can be in an increasing real sequence with average gap 1 that exhibits Poisson pair correlations (PPC). Building on prior bounds that allowed gaps up to 2 and a near $\tfrac{3}{2}$ threshold, the authors prove a sharper constraint: $\limsup_{n\to\infty}(\lambda_{n+1}-\lambda_n) > \tfrac{3}{2}+10^{-9}$. The proof combines a careful decomposition of the sequence into blocks with total gap at most $\tfrac{1}{2}$, a greedy partition to manage $0$-density regions, and a bias analysis near zero to show cross-block PPC contributions cannot be reconciled with PPC for all intervals. The result narrows the feasible range for PPC sequences, revealing deeper structure in how PPC manifests in real sequences and contributing to a broader understanding of random-like spacing phenomena in number theory.

Abstract

We prove that any increasing sequence of real numbers with average gap $1$ and Poisson pair correlations has some gap that is at least $3/2+10^{-9}$. This improves upon a result of Aistleitner, Blomer, and Radziwill.

On the smallest gap in a sequence with Poisson pair correlations

TL;DR

This work addresses how small the largest gap can be in an increasing real sequence with average gap 1 that exhibits Poisson pair correlations (PPC). Building on prior bounds that allowed gaps up to 2 and a near threshold, the authors prove a sharper constraint: . The proof combines a careful decomposition of the sequence into blocks with total gap at most , a greedy partition to manage -density regions, and a bias analysis near zero to show cross-block PPC contributions cannot be reconciled with PPC for all intervals. The result narrows the feasible range for PPC sequences, revealing deeper structure in how PPC manifests in real sequences and contributing to a broader understanding of random-like spacing phenomena in number theory.

Abstract

We prove that any increasing sequence of real numbers with average gap and Poisson pair correlations has some gap that is at least . This improves upon a result of Aistleitner, Blomer, and Radziwill.
Paper Structure (6 sections, 12 theorems, 80 equations)

This paper contains 6 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Motivation and Proof Sketch of Theorem \ref{['main']}
  3. Proof of Theorem \ref{['main']}
  4. Partitioning, and a proof of Proposition \ref{['partition']}
  5. Appendix: Proof of Lemma \ref{['5/12lemma']}
  6. Acknowledgments

Key Result

Theorem 1

Let $\lambda_1 < \lambda_2 < \dots$ be an increasing sequence of real numbers with average gap $1$ and Poisson pair correlations. Then $\limsup_{n \to \infty} \lambda_{n+1}-\lambda_n > \frac{3}{2}+10^{-9}$.

Theorems & Definitions (27)

  • Theorem 1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Definition 3.6
  • ...and 17 more