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Funktsional'nyi Analiz i ego Prilozheniya, 2015, Volume 49, Issue 2, Pages 39–53
DOI: https://doi.org/10.4213/faa3185 (Mi faa3185) 		 
This article is cited in 7 scientific papers (total in 7 papers)

A quantitative version of the Beurling-Helson theorem

S. V. Konyaginab, I. D. Shkredovca

a Steklov Mathematical Institute of Russian Academy of Sciences
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
Full-text PDF (206 kB) Citations (7)
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DOI: https://doi.org/10.4213/faa3185
Abstract: 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.
Funding agency Grant number
Russian Foundation for Basic Research 14-01-00332
12-01-33080-мол_а_вед
Ministry of Education and Science of the Russian Federation НШ-3082.2014.1
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.
Received: 14.01.2014
English version:
Functional Analysis and Its Applications, 2015, Volume 49, Issue 2, Pages 110–121
DOI: https://doi.org/10.1007/s10688-015-0093-0
Bibliographic databases:
Document Type: Article
UDC: 517.518.45
Language: Russian
Citation: S. V. Konyagin, I. D. Shkredov, “A quantitative version of the Beurling-Helson theorem”, Funktsional. Anal. i Prilozhen., 49:2 (2015), 39–53; Funct. Anal. Appl., 49:2 (2015), 110–121
Citation in format AMSBIB
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    • This publication is cited in the following 7 articles:
    1. Noga Alon, Matija Bucić, Lisa Sauermann, Dmitrii Zakharov, Or Zamir, “Essentially tight bounds for rainbow cycles in proper edge‐colourings”, Proceedings of London Math Soc, 130:4 (2025)  crossref
    2. I.D. Shkredov, “Additive dimension and the growth of sets”, Discrete Mathematics, 347:9 (2024), 114077  crossref
    3. M. R. Gabdullin, “Lower Bounds for the Wiener Norm in $\mathbb Z_p^d$”, Math. Notes, 107:4 (2020), 574–588  mathnet  crossref  crossref  mathscinet  isi  elib
    4. T. Sanders, “Bounds in cohen's idempotent theorem”, J. Fourier Anal. Appl., 26:2 (2020), 25  crossref  mathscinet  zmath  isi
    5. V. Lebedev, A. Olevskii, “Homeomorphic changes of variable and Fourier multipliers”, J. Math. Anal. Appl., 481:2 (2020), 123502  crossref  mathscinet  zmath  isi
    6. I. D. Shkredov, Trigonometric Sums and Their Applications, 2020, 261  crossref
    7. V. Lebedev, “Quantitative aspects of the Beurling-Helson theorem: phase functions of a special form”, Studia Math., 247:3 (2019), 273–283  crossref  mathscinet  zmath  isi  scopus
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Функциональный анализ и его приложения Functional Analysis and Its Applications
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