Geodesic
A quantitative version of the Beurling-Helson theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 39-53

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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.
Keywords: Wiener norm, Beurling-Helson theorem, dissociated sets. 
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     author = {S. V. Konyagin and I. D. Shkredov},
     title = {A quantitative version of the {Beurling-Helson} theorem},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     year = {2015},
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     url = {http://geodesic.mathdoc.fr/item/FAA_2015_49_2_a4/}
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S. V. Konyagin; I. D. Shkredov. A quantitative version of the Beurling-Helson theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 39-53. http://geodesic.mathdoc.fr/item/FAA_2015_49_2_a4/