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arXiv · 2109.05379

Additive energy, discrepancy and Poissonian $k$-level correlation

Guy Lachman·Shvo Regavim
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Abstract

$k$-level correlation is a local statistic of sequences modulo 1, describing the local spacings of $k$-tuples of elements. For $k = 2$ this is also known as pair correlation. We show that there exists a well spaced increasing sequence of reals with additive energy of order $N^3$ and Poissonian $k$-level correlation for all integers $k \geq 2$, answering in the affirmative a question raised by Aistleitner, El-Baz, and Munsch. The construction is probabilistic, and so we do not obtain a specific sequence satisfying this condition. To prove this, we show that random perturbations of a sequence with small discrepancy gives, almost surely, a sequence with Poissonian $k$-level correlation, a fact which may be of independent interest.

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Guy Lachman, Shvo Regavim. 2021-09-11. Additive energy, discrepancy and Poissonian $k$-level correlation. https://arxiv.org/abs/2109.05379

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