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We exhibit a class of Littlewood polynomials that are not $L^\alpha$-flat for any $\alpha \geq 0$. Indeed, it is shown that the sequence of Littlewood polynomials is not $L^\alpha$-flat, $\alpha \geq 0$, when the frequency of $-1$ is not in the interval $]\frac14,\frac34[$. We further obtain a generalization of Jensen-Jensen-Hoholdt's result by establishing that the sequence of Littlewood polynomials is not $L^\alpha$-flat for any $\alpha> 2$ if the frequency of $-1$ is not $\frac12$. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not $L^\alpha$-flat for any $\alpha \geq 0$., Comment: 15 pages. The appendix is written jointly with M, G. Nadkarni. In this new version few misprints are corrected and Jensen-Jensen-Hoholdt's theorem is generalized

We formulate and prove a necessary condition for a sequence of analytic trigonometric polynomials with real non-negative coefficients to be flat a.e., Comment: In this paper we continue the investigation started in "Some Notes on Flat Polynomials" [arXiv:1402.5457]

Connection of flat polynomials with some spectral questions in ergodic theory is discussed. A necessary condition for a sequence of polynomials of the type $\frac{1}{\sqrt{N}} \big(1 +\sum_{j=1}^{N-1} z^{n_j}\big)$ to be flat in almost everywhere sense is given, which contrasts with a similar necessary condition for a sequence of polynomials to be ultraflat., Comment: 13 pages

In this note we give an example of an ergodic non-singular map whose unitary operator admits a Lebesgue component of multiplicity one in its spectrum., Comment: The paper is accepted for publication in ETDS, and in this revised version, we gives a short proof of a new Lemma of L^1-flatness that state that if the spectral type of generalized odometer (rank one maps in particular case) has the same null sets as the Lebesgue measure then the associated sequence of polynomials is L^1-flat

In this paper we discuss generalized Riesz products bringing into consideration $H^p$ theory, the notion of Mahler measure, and the zeros of polynomials appearing in the generalized Riesz product. Formula for Radon-Nikodym derivative between two generalized Riesz product is established under suitable conditions. This is then used to formulate a Dichotomy theorem and prove a conditional version of it. A discussion involving flat polynomials is given., Comment: 30 pages, accepted for publication

In this paper we give an improved upper bound, as compared to the one given in [3] for the number of extreme points of the convex set of all G-invariant probability measures on X*Y with given marginals of full support., Comment: 11 pages

We give a necessary and sufficient condition on a sequence of functions on a set $\Omega$ under which there is a measure on $\Omega$ which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities., Comment: 6 pages

We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras., Comment: 17 pages, no figures, no tables

We discuss subsets $S$ of ${\mathbb{R}}^n$ such that every real valued function $f$ on $S$ is of the form {equation*} f(x_1,x_2,...,x_n) = u_1(x_1) + u_2(x_2) + ... + u_n(x_n), {equation*} and the related concepts and situations in analysis., Comment: 10 pages, no figures, no tables