Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem

Abstract: We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}={\hat{f}(k), ~k \in \mathbb Z}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $|f|{A(\mathbb T)}=|\hat{f}|{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $\phi : \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $|e^{in\phi}|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $\phi$ is linear. It was conjectured by Kahane that the same conclusion about $\phi$ is true under the assumption that $|e^{in\phi}|_{A(\mathbb T)}=o(\log |n|)$. We show that if $|e^{in\phi}|_{A(\mathbb T)}=o((\log\log |n|/\log\log\log |n|)^{1/12})$ then $\phi$ is linear.