The quantitative Beurling-Helson Theorem

Abstract

We show that for any $\epsilon >0$ if $\ddot{I}•:\mathbb{T}\to \mathbb{T}$ is continuous and $\Vert \exp (-2\ddot{I}€iz\ddot{I}•){\Vert }_{A(\mathbb{T})}=O_{|z|\to \infty }({\log }^{\frac{1}{8}-\epsilon }|z|)$ then $\ddot{I}•(x)=wx+t$ for some $w\in \mathbb{Z}$ and $t\in \mathbb{T}$.

16pp