Organizer: Shaoyun Yi

Email: yishaoyun926@xmu.edu.cn

Office: B515

Tentative Schedule:

Week No.	Date	Topics	Speaker
1	2/19 (W)	The metric theory of the pair correlation function for small non-integer powers (Abstract)	Lin Feng
2	2/25	Some interesting combinatorics problems	Boyi Zheng
3	3/4	Some interesting techniques in integral calculus with an application: Proving the transcendence of $e$     An introduction to Zermelo-Fraenkel set theory I	Chenxi Guan     Rui Yan
4	3/11	An introduction to Zermelo-Fraenkel set theory II: Natural numbers	Rui Yan
5	3/18	An introduction to Zermelo-Fraenkel set theory III: A linear ordering of natural numbers	Rui Yan
6	3/25	An introduction to Zermelo-Fraenkel set theory IV: The (first) recursion theorem	Rui Yan
7	4/1	Banach fixed-point theorem and the norm equivalence theorem in finite-dimensional spaces	Junjie Jiang
8	4/8
9	4/15
10	4/22
11	4/29
12	5/6
13	5/13
14	5/20
15	5/27
16	6/3

Abstracts

Lin Feng - The metric theory of the pair correlation function for small non-integer powers

The pair correlation is a localized statistic for sequences in the unit interval. The metric theory of pair correlations of sequences of the form $(\alpha a_n)_{n\ge 1}$ has been pioneered by Rudnick, Sarnak, and Zaharescu in 2001. Here $\alpha$ is a real parameter and $(a_n)_{n\ge 1}$ is an integer sequence, often of arithmetic origin. In particular, we study if for almost all $\alpha$, the pair correlation of the sequence $x_n=\langle \alpha n^{\theta}\rangle, n\ge 1$ is Poissonian, i.e. \[ \lim_{N\to\infty}\frac{1}{N} \#\{1\le n\neq m\le N: |x_n-x_m|\le s/N\}=2s \] for all $s\ge 0$.
In 1998, Rudnick and Sarnak proved the corresponding result for integer $\theta>1$.
In 2021, Aistleitner, El-Baz and Munsch obtained some criteria for real valued sequences $(a_n)_{n\ge 1}$, which could solve the case of non-integer $\theta>1$.
In 2022, Rudnick and Technau completed the problem by solving the final case when $\theta\in(0,1)$.
In this talk, we will focus on Rudnick and Technau's result, where they used Dirichlet polyomial to do decoupling.