Littlewood polynomials and applications of them in the spectral theory of dynamical systems

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 $\mathbb R$.
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