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A quantitative version of the Beurling-Helson theorem. (English. Russian original) Zbl 1326.42005

Funct. Anal. Appl. 49, No. 2, 110-121 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 39-53 (2015).
Summary: 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 a prime number \(p\) are obtained.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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References:

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