[Paper Review] Remark on a result of Bourgain on poissonian pair correlation

Gerhard Larcher|arXiv (Cornell University)|Nov 23, 2017

Limits and Structures in Graph Theory10 references3 citations

TL;DR

This paper proves that for a broad class of integer sequences $(a_n)$ with positive upper density—such as strictly increasing sequences—the sequence $\{a_n\alpha\}$ fails to have Poissonian pair correlation for any irrational $\alpha$. The result supports a conjecture that such pair correlation fails whenever the additive energy $E(A_N) = \Omega(N^3)$, extending Bourgain's negative result to all $\alpha$, not just almost all.

ABSTRACT

We show for a class of sequences $(a_n)_{n\geq 1}$ of distinct positive integers, that for no $α$ the sequence $(\left\{a_n α ight\})_{n \geq 1}$ does have Poissonian pair correlation. This class contains for example all strictly increasing integer sequences with positive upper density. This result motivates us to state a certain conjecture on Poissonian pair correlation which would be a significantly stronger version of a result of Jean Bourgain.

Motivation & Objective

• To investigate whether the absence of Poissonian pair correlation in sequences $\{a_n\alpha\}$, previously known for almost all $\alpha$, holds for all $\alpha$ when additive energy $E(A_N) = \Omega(N^3)$.
• To extend Bourgain's result—which showed non-Poissonian correlation for a positive measure set of $\alpha$—to the full set of $\alpha$ under stronger structural conditions on $a_n$.
• To provide evidence for a conjecture that $E(A_N) = \Omega(N^3)$ implies no $\alpha$ yields Poissonian pair correlation.
• To analyze sequences with strong linear substructure (via Balog–Szemerédi–Gowers and Freiman's theorems) to derive quantitative lower bounds on pair correlation functions.
• To demonstrate that for such sequences, the pair correlation function $R_N(s)$ cannot converge to $2s$, contradicting the Poissonian condition.

Proposed method

• Uses the equivalence $E(A_N) = \Omega(N^3) \iff \sum_v A_N^2(v) = \Omega(N^3)$, where $A_N(v)$ counts differences $a_i - a_j = v$.
• Applies the Balog–Szemerédi–Gowers theorem to extract a large subset $A_0^{(i)} \subset \{a_n\}_{n \leq N_i}$ with small doubling: $|A_0^{(i)} + A_0^{(i)}| \leq C|A_0^{(i)}|$.
• Applies Freiman's theorem to show that such a set is contained in a $d$-dimensional arithmetic progression of size $O(N_i)$, implying linear structure.
• Constructs a finite set $\mathcal{E}$ of $s$-values depending only on universal constants $c, K$ to test pair correlation at critical scales.
• Analyzes the pair correlation function $R_N(s) = \frac{1}{N}\#\{i \neq j : \|x_i - x_j\| \leq s/N\}$ and derives lower bounds that violate convergence to $2s$.
• Uses discrepancy arguments and interval counting in dyadic partitions to show that $R_N(s_2) - R_N(s_1) \geq 4(s_2 - s_1)$ for some $s_1, s_2$, contradicting Poissonian behavior.

Experimental results

Research questions

• RQ1 Does $E(A_N) = \Omega(N^3)$ imply that $\{a_n\alpha\}$ fails to have Poissonian pair correlation for every $\alpha$, not just almost every $\alpha$?
• RQ2 Can the class of sequences with positive upper density be shown to lack Poissonian pair correlation for all $\alpha$?
• RQ3 What structural properties (e.g., additive energy, doubling constants) are sufficient to prevent Poissonian pair correlation?
• RQ4 Can quantitative lower bounds on $R_N(s)$ be derived to contradict the Poissonian limit $2s$?
• RQ5 Is there a finite set of $s$-values (depending only on structural parameters) such that $R_N(s)$ cannot converge to $2s$?

Key findings

• For any sequence $\{a_n\}$ of distinct positive integers with positive upper density, $\{a_n\alpha\}$ fails to have Poissonian pair correlation for any irrational $\alpha$.
• The result holds for all $\alpha$, not just almost all, thus strengthening Bourgain's earlier result which required exceptional $\alpha$ of positive measure.
• The proof relies on the existence of a large subset $A_0^{(i)} \subset \{a_n\}_{n \leq N_i}$ with $|A_0^{(i)}| \geq cN_i$ and $|A_0^{(i)} + A_0^{(i)}| \leq C|A_0^{(i)}|$, implying linear structure via Freiman's theorem.
• A finite set $\mathcal{E}$ of $s$-values is constructed such that for some $s_1, s_2 \in \mathcal{E}$, the difference $R_N(s_2) - R_N(s_1) \geq 4(s_2 - s_1)$, violating the Poissonian limit $2(s_2 - s_1)$.
• The pair correlation function $R_N(s)$ cannot converge to $2s$ for certain $s$, including $s = \frac{2^6}{c}$, $s = 1$, or $s \in \mathcal{D}$ or $\mathcal{E}$, due to persistent discrepancy.
• The lower bound $R_N(s) \geq \frac{c^5}{2^{24}K^2}b > 4$ is derived for some $s$, contradicting the expected $2s$ limit, thus proving non-convergence.