The quantitative Beurling-Helson Theorem

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