Long-Range Correlations of Sequences Modulo 1

TL;DR

This paper develops a method to analyze long-range correlations of fractional parts of sequences like lpha \,sqrt{n} and lpha n^2, showing they are Poissonian and indicating pseudo-random behavior.

Contribution

It introduces a general approach linking correlation convergence to bounds on Weyl sums, extending previous results to higher moments and sequences.

Findings

Long-range correlations are Poissonian for the sequences studied.

The methodology recovers and extends previous results on triple correlations.

Higher moments of correlations suggest pseudo-randomness.

Abstract

In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations.