A quantitative version of the Beurling-Helson theorem

It is proved that any continuous function $\varphi$ on the unit circle such that the sequence $\{e^{in\varphi}\}_{n\in\mathbb{Z}}$ has small Wiener norm $\|e^{in\varphi}\| = o(\log^{1/22}|n|(\log \log |n|)^{-3/11})$, $|n| \to \infty$, is linear. Moreover, lower bounds for the Wiener norms of the characteristic functions of subsets of $\mathbb{Z}_p$ in the case of prime $p$ are obtained.