Geodesic
Littlewood polynomials and applications of them in the spectral theory of dynamical systems
Sbornik. Mathematics, Tome 204 (2013) no. 6, pp. 910-935

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In this paper we establish the existence of character sums on the real line $\mathbb R$ that are $\varepsilon$-flat on any given compact subset $K\subset \mathbb R \setminus \{0\}$ with respect to the metric in the space $L^1(K)$. A consequence of this analytic result is an affirmative answer to Banach's conjecture on the existence of a dynamical system with a simple Lebesgue spectrum in the class of actions of the group $\mathbb R$. Bibliography: 25 titles.
Keywords: Littlewood polynomials, van der Corput's method, Riesz products, rank-one flows, Banach's problem. 
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     title = {Littlewood polynomials and applications of them in the spectral theory of dynamical systems},
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A. A. Prikhod'ko. Littlewood polynomials and applications of them in the spectral theory of dynamical systems. Sbornik. Mathematics, Tome 204 (2013) no. 6, pp. 910-935. http://geodesic.mathdoc.fr/item/SM_2013_204_6_a4/