The distribution of spacings between the fractional parts of $n^2 \alpha$

TL;DR

This paper investigates the distribution of spacings between fractional parts of quadratic sequences, proposing a conjecture relating Diophantine properties of a to Poissonian spacings, and providing partial results and counterexamples.

Contribution

It introduces a conjecture linking Diophantine approximation properties of a to Poisson spacings and offers partial proofs and a counterexample for higher correlation functions.

Findings

Conjecture that badly approximable a leads to Poisson spacings.

Partial results supporting the conjecture.

Counterexample showing higher correlations can diverge.

Abstract

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.