Let $\mathcal{A}\left(\alpha\right)$ denote the sequence $\left(\alpha a_{n}\right)_{n}$, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of these sequences follows the Poissonian model for almost all $\alpha$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker were even led to speculate that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.

中文翻译：

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没有度量对相关性的 KHINTCHINE 阈值

让 $\mathcal{A}\left(\alpha\right)$ 表示序列 $\left(\alpha a_{n}\right)_{n}$，其中 $\alpha\in\left[0,1 \right]$ 并且其中 $\left(a_{n}\right)_{n}$ 是一个严格递增的正整数序列。如果这些序列的对相关性的渐近分布在 Lebesgue 测度意义上对几乎所有 $\alpha$ 都遵循 Poissonian 模型，我们说 $(a_n)_n$ 具有度量对相关性。最近的研究揭示了此类序列对相关性的度量理论与 $(a_n)_{n}$ 截断的附加能量之间的联系。Bloom、Chow、Gafni 和 Walker 甚至推测可能存在一个收敛/发散标准，它完全表征了附加能量方面的度量对相关属性，类似于 Khintchine' 丢番图近似度量理论中的 s 准则。在本文中，我们通过表明不存在这样的标准来对这种推测给出否定的答案。为此，我们构建了一个具有大附加能量的序列 $(a_n)_n$，然而，它保持了度量对相关性。