Pair correlation of monomial sequences modulo 1

Abstract: Fix $\alpha, \theta > 0$, and consider the sequence $(\alpha n^\theta \mod 1)_{n>0}$. Since the seminal work of Rudnick-Sarnak (1998), and due to the Berry-Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this talk, I will briefly survey what is known about these sequences and present a recent result (joint with Sourmelidis and Technau) showing that for $\theta \le 1/3$, and $\alpha > 0$, the pair correlation function is Poissonian. While the techniques we use are derived from analytic number theory, the problem is rooted in dynamics and relates to dynamical proofs for related problems.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar