The distribution of spacings between the fractional parts of ${\mathbf{n}}^{\mathbf{d}}\hat{\mathbf{I}}\pm$

Abstract

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

23 pages