Littlewood polynomials and applications of them in the spectral theory of dynamical systems (Q2848653)

The trigonometric sums NEWLINE\[NEWLINE \mathcal{P}_{\omega,q}(t)=\frac{1}{\sqrt{q}}\sum_{y=0}^{q-1}e^{2\pi it\omega(y)},\; t,\omega(y)\in \mathbb{R} NEWLINE\]NEWLINE are called \(\varepsilon\)-flat in the space \(L^1([t_1,t_2])\) if NEWLINE\[NEWLINE \left\| |\mathcal{P}_{\omega,q}(t)|-1 \right\|_{1}\leq\varepsilonNEWLINE\]NEWLINE The main result of the paper is that for every sufficiently large \(m\) there exist \(\beta\) and an infinite sequence of \(q_j\) such that \(\mathcal{P}_{\omega,q_j}(t)\) are \(\varepsilon\)-flat with \(\omega(y)=m\frac{q_j}{\beta^2}e^{\beta y/q_j}.\) The main idea of the proof is a double application of the precise version, due to \textit{H. Liu} [Acta Arith. 90, No. 4, 357--370 (1999; Zbl 0934.11039)], of the van der Corput's method. After a second application of the method, a sum consisting of only one summand is obtained. This is the main step of the proof and it is the choice of the function \(\omega(y)\) that allows to obtain such a short sum. The author states: ``A consequence of the main 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 on the group \(\mathbb{R}\)''.