The quantitative Beurling-Helson Theorem

Abstract

We show that for any $\varepsilon>0$ if $\phi:\mathbb{T} \rightarrow \mathbb{T}$ is continuous and $\|\exp(-2\pi i z \phi)\|_{A(\mathbb{T})} =O_{|z|\rightarrow \infty}(\log^{\frac{1}{8}-\varepsilon} |z|)$ then $\phi(x)=wx+t$ for some $w \in\mathbb{Z}$ and $t \in \mathbb{T}$.