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.
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.
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