Absolutely Convergent Fourier Series. An Improvement of the Beurling–Helson Theorem

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