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Full Poissonian Local Statistics of Slowly Growing Sequences

Christopher Lutsko, Niclas Technau

TL;DR

This work proves that the deterministic sequence $x(n)=\alpha(\log n)^A\bmod 1$ with $A>1$ has Poissonian local statistics: every $m$-point correlation $R^{(m)}(N,f)$ converges to the Poisson limit for all $m\ge 2$, implying a Poisson gap distribution. The authors develop a multi-scale analysis combining a completed-moment reduction, a two-variable van der Corput $B$-process, and a diagonal/off-diagonal extraction; they also leverage stationary-phase arguments and Khare--Tao bounds on generalized Vandermonde determinants. The threshold $A>1$ is optimal in light of Marklof--Strömbergsson's results for logarithmic growth, and this work provides the first explicit deterministic example with Poissonian correlations of all orders. The techniques offer a blueprint for establishing fine-scale statistics for sub-polynomial growth sequences and highlight the role of combinatorial partitions and oscillatory integral bounds in obtaining Poisson limits.

Abstract

Fix $α>0$, then by Fejér's theorem $ (α(\log n)^{A}\,\mathrm{mod}\,1)_{n\geq1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided $A>1$. This is the first example of a deterministic sequence modulo one whose gap distribution, and all of whose correlations are proven to be Poissonian. The range of $A$ is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of $(\log(n)\, \mathrm{mod}\,1)$, which is necessarily not Poissonian.

Full Poissonian Local Statistics of Slowly Growing Sequences

TL;DR

This work proves that the deterministic sequence with has Poissonian local statistics: every -point correlation converges to the Poisson limit for all , implying a Poisson gap distribution. The authors develop a multi-scale analysis combining a completed-moment reduction, a two-variable van der Corput -process, and a diagonal/off-diagonal extraction; they also leverage stationary-phase arguments and Khare--Tao bounds on generalized Vandermonde determinants. The threshold is optimal in light of Marklof--Strömbergsson's results for logarithmic growth, and this work provides the first explicit deterministic example with Poissonian correlations of all orders. The techniques offer a blueprint for establishing fine-scale statistics for sub-polynomial growth sequences and highlight the role of combinatorial partitions and oscillatory integral bounds in obtaining Poisson limits.

Abstract

Fix , then by Fejér's theorem is uniformly distributed if and only if . We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided . This is the first example of a deterministic sequence modulo one whose gap distribution, and all of whose correlations are proven to be Poissonian. The range of is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of , which is necessarily not Poissonian.
Paper Structure (14 sections, 19 theorems, 206 equations, 1 figure)

This paper contains 14 sections, 19 theorems, 206 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Plan of Paper
  3. Preliminaries
  4. Combinatorial Completion
  5. Combinatorial Target
  6. Dyadic Decomposition
  7. Applying the $B$-process
  8. Degenerate Regimes
  9. First application of the $B$-Process
  10. Amplitude Bounds
  11. Second Application of the $B$-Process
  12. Extracting the Diagonal
  13. Bounding the Off-Diagonal
  14. Proof of Lemma \ref{['lem:MP = KP non-isolating']}

Key Result

Theorem 1.1

Let $\omega(n):= \alpha ( \log n)^{A}$ for $A>1$ and any $\alpha >0$, then $x(n)$ has Poissonian gap distribution.

Figures (1)

  • Figure 1: From left to right: the histograms represent the gap distribution density at time $N$ of $(\log n)_{n\geq1}$, $((\log n)^{2})_{n > 0}$, and $((\log n)^{3})_{n> 0}$ when $N=10^{5}$ and the curve is the graph of $x\mapsto e^{-x}$. Note that $(\log n)$ is not even uniformly distributed, and thus the gap distribution cannot be Poissonian.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2: First order stationary phase
  • proof
  • Lemma 2.3: van der Corput's lemma
  • Lemma 2.4: KhareTao2021
  • Lemma 3.1
  • Remark
  • Lemma 4.1
  • ...and 23 more