A new theorem shows that a small-scale additive energy estimate plus a weighted large-gap sum implies metric Poissonian pair correlation, yielding the property for quantitatively convex and real polynomial sequences.
Metric Poissonian pair correlationa and additive energy
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abstract
In this article we prove that for a strictly increasing sequence $(a_n)$ of natural numbers, if the additive energy of $\{a_n:n\leq N\}$ is less than $N^3/(\log N)^C$ for some $C\geq14.71,$ then $(\{a_n\alpha\})$ has Poissonian pair correlation for almost all $\alpha\in\mathbb{R}.$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker [4].
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Metric Poissonian pair correlation for real sequences and energy estimates
A new theorem shows that a small-scale additive energy estimate plus a weighted large-gap sum implies metric Poissonian pair correlation, yielding the property for quantitatively convex and real polynomial sequences.