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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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 49, No. 2, pp. 39–53, 2015
Original Russian Text Copyright © by S. V. Konyagin and I. D. Shkredov
The first author acknowledges the support of RFBR grant no. 14-01-00332 and of the program “Leading Scientific Schools,” grant no. 3082.2014.1. The second author acknowledges the support of RFBR grant no. 12-01-33080-mol a ved.
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Konyagin, S.V., Shkredov, I.D. A quantitative version of the Beurling-Helson theorem. Funct Anal Its Appl 49, 110–121 (2015). https://doi.org/10.1007/s10688-015-0093-0
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DOI: https://doi.org/10.1007/s10688-015-0093-0