A Class of Littlewood Polynomials that are Not Lα-Flat

We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not Lα-flat, α ≥ 0, when the frequency of −1 is not in the interval ]{1 \over 4}, {3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not Lα-flat for any α> 2 if the frequency of −1 is not {1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not Lα-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

Morse cocycles and simple Lebesgue spectrum

It is well known that the spectrum of dynamical systems arising from generalized Morse sequences (M. Keane. Generalized Morse sequences. Z. Wahr. Verw. Geb.10 (1968), 335–353.) is simple as soon as it is non-discrete. We give, in this paper, a necessary and sufficient condition for the existence of such a transformation with non-purely singular spectrum. From this, it follows that this problem is equivalent to an open problem of the existence of ‘flat’ polynomials on the Torus group. We show that this latter question can be given an affirmative answer on some other group, and this allows us to construct a countable abelian group action with simple spectrum whose spectral type is the sum of a discrete measure and of the Haar measure on the dual group.