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arXiv · 2112.11524

Correlations of the Fractional Parts of $\alpha n^\theta$

Christopher Lutsko·Niclas Technau
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Abstract

Let $m\geq 3$, we prove that $(\alpha n^\theta \mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $\alpha>0$, provided $\theta<\theta_m$, where $\theta_m$ is an explicit bound which goes to $0$ as $m$ increases. This work builds on the method developed in Lutsko-Sourmelidis-Technau (2021), and introduces a new combinatorial argument for higher correlation levels, and new Fourier analytic techniques. A key point is to introduce an `extra' frequency variable to de-correlate the sequence variables and to eventually exploit a repulsion principle for oscillatory integrals. Presently, this is the only positive result showing that the $m$-point correlation is Poissonian for such sequences.

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Christopher Lutsko, Niclas Technau. 2021-12-21. Correlations of the Fractional Parts of $\alpha n^\theta$. https://arxiv.org/abs/2112.11524

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