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Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in {\mathbb N}}$ with maximal additive energy



We show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.



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Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in {\mathbb N}}$ with maximal additive energy



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