The distribution of spacings between the fractional parts of $\boldsymbol{n^d\alpha}$

TL;DR

This paper investigates the distribution of spacings between fractional parts of polynomial sequences and establishes conditions under which these spacings follow a Poisson distribution for certain algebraic irrationals.

Contribution

It provides a necessary and sufficient condition for the spacings to be Poissonian for high Diophantine type irrationals, advancing understanding of fractional parts distribution.

Findings

Identifies conditions for Poissonian spacings in polynomial fractional parts

Proves the result for irrationals of high Diophantine type

Establishes a link between Diophantine approximation and spacings distribution

Abstract

We study the distribution of spacings between the fractional parts of $n^d\alpha$. For $\alpha$ of high enough Diophantine type we prove a necessary and sufficient condition for $n^d\alpha\mod 1, 1\leq n\leq N,$ to be Poissonian as $N\to \infty$ along a suitable subsequence.