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