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e. H. el Abdalaoui

Publications and source records attributed to e. H. el Abdalaoui.

6 recordsLinked to original sources
arXiv2022

Möbius random law and infinite rank-one maps

We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the Möbius function is a good weight for the ergodic theorem, to maps acting on $σ$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the Möbius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on Möbius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.

math.DS↗
arXiv2021

Weakly tame systems, their characterizations and application

We explore the notion of discrete spectrum and its various characterizations for ergodic measure-preserving actions of an amenable group on a compact metric space. We introduce a notion of 'weak-tameness', which is a measure-theoretic version of a notion of `tameness' introduced by E. Glasner, based on the work of A. Köhler [A. Köhler introduced this notion and call such systems "regular".], and characterize such topological dynamical systems as systems for which every invariant measure has a discrete spectrum. Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of 'witness of irregularity' which is based on up-crossings. Then we establish that strong Veech systems are tame. In particular, for any amenable group $T$, the flow on the orbit closure of the translates of a `Veech function' $f\in \mathbb{K}(T)$ is tame. Thus Sarnak's Möbius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi-Ramachandra 1976's theorem on the Mertens function in short intervals. We further improve Motohashi-Ramachandra's bound to $1/2$ under Chowla conjecture.

math.DS↗
arXiv2019

Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a consequence, we establish that the homogeneous bilinear ergodic averages along polynomials and polynomials in primes converge almost everywhere, that is, for any invertible measure preserving transformation $T$, acting on a probability space $(X, \mathcal{B}, μ)$, for any $f \in L^r(X,μ)$ , $g \in L^{r'}(X,μ)$ such that $\frac{1}{r}+\frac{1}{r'}= 1$, for any non-constant polynomials $P(n),Q(n), n \in \mathbb{Z}$, taking integer values, and for almost all $x \in X$, we have, $$\frac{1}{N}\sum_{n=1}^{N}f(T^{P(n)}x) g(T^{Q(n)}x),$$ and $$\frac{1}{π_N}\sum_{\overset{p \leq N}{p\textrm{~~prime}}}f(T^{P(p)}x) g(T^{Q(p)}x),$$ converge. Here $π_N$ is the number of prime in $[1,N]$.

math.DS↗
arXiv2014

Some Notes on Flat Polynomials

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.

math.CV↗
arXiv2013

Calculus of Generalized Riesz Products

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.

math.DS↗