On the correlations of $n^\alpha$ mod 1

Niclas Technau; Nadav Yesha

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arXiv · 2006.16629

On the correlations of $n^\alpha$ mod 1

Niclas Technau·Nadav Yesha

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Abstract

A well known result in the theory of uniform distribution modulo one (which goes back to Fej\'er and Csillag) states that the fractional parts $\{n^\alpha\}$ of the sequence $(n^\alpha)_{n\ge1}$ are uniformly distributed in the unit interval whenever $\alpha>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^\alpha\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $\alpha>4k^2-4k-1$.

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Niclas Technau, Nadav Yesha. 2020-06-30. On the correlations of $n^\alpha$ mod 1. https://arxiv.org/abs/2006.16629

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