<p>In this note, we propose to study the rate at which the weak pair correlation statistic with parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0 < \alpha \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> converges to its limit in the Poissonian case. As an application we show that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(x_n)_{n \in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> has weak <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Poissonian pair correlations and the rate of convergence is locally uniform of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$cN^{-\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mi>κ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, then it also has weak <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-Poissonian pair correlations for some <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta > \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> depending on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>. This may be regarded as a partial converse of the well-known fact that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-Poissonian pair correlations imply <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Poissonian pair correlations if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\beta > \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the convergence rate of pair correlations

期刊论文
出版模式：

开放获取
发表日期：

2026年1月4日

查看全文

Christian Weiss

摘要

In this note, we propose to study the rate at which the weak pair correlation statistic with parameter $0 < \alpha \le 1$ $0 < α \leq 1$ converges to its limit in the Poissonian case. As an application we show that if $(x_n)_{n \in \mathbb {N}}$ ${(x_{n})}_{n \in N}$ has weak $\alpha $ $α$ -Poissonian pair correlations and the rate of convergence is locally uniform of order $cN^{-\kappa }$ $c N^{- κ}$ , then it also has weak $\beta $ $β$ -Poissonian pair correlations for some $\beta > \alpha $ $β > α$ depending on $\kappa $ $κ$ . This may be regarded as a partial converse of the well-known fact that $\beta $ $β$ -Poissonian pair correlations imply $\alpha $ $α$ -Poissonian pair correlations if $\beta > \alpha $ $β > α$ .