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 $\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\}$ belongs to l ¹(ℤ). The norm on $A(\mathbb{T})$ is defined by $\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}$. According to the well-known Beurling-Helson theorem, if $\phi :\mathbb{T} \to \mathbb{T}$ is a continuous mapping such that $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1)$, n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right)$. We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$, then φ is linear.