Speaker: Svetlana Jitomirskaya
16 December, 2020 16:00 UTC
Abstract: Given $n\in\mathbb{N}$ and $x\in\mathbb{R}$, let \(||nx||^\prime=\min\{|nx-m|:m\in\mathbb{Z}, gcd(n,m)=1\}.\) Two conjectures in the coprime inhomogeneous Diophantine approximation stated, by analogy with the classical Diophantine approximation, that for any irrational number $\alpha$ and almost every $\gamma\in \mathbb{R}$, \(\liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime}=0,\) and that there exists $C$ such that for all $\gamma\in \mathbb{R}$,
\[\liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime} < C.\]We will present our joint work with W. Liu that proves one of those and disproves the other.