# On the correlations of $n_{α}$ mod 1

### Niclas Technau

Tel Aviv University, Israel### Nadav Yesha

University of Haifa, Israel

## Abstract

A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts ${n_{α}}$ of the sequence $(n_{α})_{n≥1}$ are uniformly distributed in the unit interval whenever $α>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $({n_{α}})_{n≥1}$ are of fundamental importance. We prove that for each $k≥2,$ the $k$-level correlation function $R_{k}$ is Poissonian for almost every $α>4k_{2}−4k−1$.

## Cite this article

Niclas Technau, Nadav Yesha, On the correlations of $n_{α}$ mod 1. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 4123–4154

DOI 10.4171/JEMS/1281