## EVENT DETAILS AND ABSTRACT

**Number Theory**
**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.

**Date:** Friday, April 29, 2022

**Time:** 3:00PM

**Where:** Lunt 107

**Contact Person:** Ilya Khayutin

**Contact email:** khayutin@northwestern.edu

**Contact Phone:**

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