EVENT DETAILS AND ABSTRACT

Title: Distribution of spacings of the fractional parts of quadratic sequences
Speaker: Thomas Hille
Speaker Info: Northwestern University
Brief Description:
Special Note:
Abstract:

The pair correlation density for a sequence of $N$ numbers $\theta_1, \dots, \theta_N$ in $[0,1]$ measures the distribution of spacings between the elements $\theta_n$ of the sequence at distances of order of the mean spacing $N^{-1}$. The sequence of fractional parts $\{ n^2 \alpha \}_n$ has been of special interest due to its connection to a conjecture of Berry and Tabor on the energy levels of generic completely integrable systems. However, only metric results are known as of now. In this talk, I will study the distribution of spacings of such a sequence at distances of order of $N^\sigma$ ($0 \leq \sigma <2$); this point of view goes back to Nair and Pollicott and the case $\sigma =1$ is usually referred to as Poissonian pair correlation. For $\sigma <1$ we obtain the “limiting case” with the conjectured Diophantine type for $\alpha$ and for $\sigma >1$ we obtain a metric result.