60 citations · 97 across the 30 of their papers we have counts for
35 papers
OpenAI partially disproves Littlewood's 1961 -flat theorem
el Houcein el Abdalaoui
We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on -flat polynomials. The theorem asserts that no -flat polynomials exist within the class o…
On the spectral analysis of dynamical Möbius-Sarnak process and topological entropy of Möbius fonction
el Houcein el Abdalaoui
By extending the Rokhlin-Sinai machinery relating to the entropy and countable Lebesgue component in the spectrum, we establish that the dynamical Möbius-Sarnak process has a count…
Operator ergodic theorems with Möbius "weights"
El Houcein El Abdalaoui, Michael Lin
Motivated by Sarnak's conjecture in topological dynamics for the Möbius function , we study, for a power-bounded on a Banach space , the weak convergence $$ (*) \qquad \q…
Mean ergodic theorems in and ,
el Houcein el Abdalaoui, Michael Lin
Let be the Koopman operator of a measure preserving transformation of a probability space . We study the convergence properties of the averages $M_nf:=\frac1n\sum_…
-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof
el Houcein el Abdalaoui
We present a simple proof on the existence of -flat analytic polynomials with coefficients on the circle and on the real line and we give an example of a conservative er…
Mixing sequences for non-mixing transformations and group actions
el Houcein el Abdalaoui, Terry Adams
We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences which satisfy the Rajchman dissociated property. Our examples are b…
On the Mahler measure of the spectrum of rank one maps
el Houcein el Abdalaoui
We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-product type measures by proving that the generalized Riesz-product is singular if and on…
On the homogeneous ergodic bilinear averages with -bounded multiplicative weights
el Houcein el Abdalaoui
We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic -bounded multiplicative function $\bo…
Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow
el Houcein el Abdalaoui, Mahesh Nerurkar
It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is acco…
Weakly tame systems, their characterizations and application
e. H. el Abdalaoui, M. Nerurkar
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 n…
Exponential sums of the Möbius function and flat polynomials
el Houcein el Abdalaoui
We present an elementary problem on analytic polynomials with coefficients or in which implies Riemann hypothesis. It is turns out that this problem is a par…
Disjointness of the Möbius Transformation and Möbius Function
El Houcein El Abdalaoui, Igor E. Shparlinski
We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation over a finite field of $…
On Veech's proof of Sarnak's theorem on the Möbius flow
el Houcein el Abdalaoui
We present Veech's proof of Sarnak's theorem on the Möbius flow which say that there is a unique admissible measure on the Möbius flow. As a consequence, we obtain that Sarnak's co…
On the homogeneous ergodic bilinear averages with Möbius and liouville weights
El Houcein El Abdalaoui
It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero, that is, if is a map acting on a probability space $(…
Oscillating sequences, Gowers norms and Sarnak's conjecture
el Houcein el Abdalaoui
It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that an…
Möbius disjointness for models of an ergodic system and beyond
El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk +1
Given a topological dynamical system and an arithmetic function , we study the strong MOMO property (relatively to $\boldsymbol…
On the Erdös flat polynomials problem, Chowla conjecture and Riemann Hypothesis
el Houcein el Abdalaoui
There are no square -flat sequences of polynomials of the type where for each $j,~~ 0 \leq…
A class of Littlewood polynomials that are not -flat
E. H. el Abdalaoui, M. G. Nadkarni
We exhibit a class of Littlewood polynomials that are not -flat for any . Indeed, it is shown that the sequence of Littlewood polynomials is not -flat, …
A cubic nonconventional ergodic average with multiplicative or Mangoldt weights
el Houcein el Abdalaoui, XiangDong Ye
We show that the cubic nonconventional ergodic averages of any order with a bounded multiplicative function weight converge almost surely to zero provided that the multiplicative f…
A simple proof of Bourgain's theorem on the singularity of the spectrum of Ornstein's maps
El Houcein El Abdalaoui
We give a simple proof of Bourgain's theorem on the singularity of Ornstein's maps.
On flat polynomials with non-Negative coefficients
e. H. el Abdalaoui, M. G. Nadkarni
We formulate and prove a necessary condition for a sequence of analytic trigonometric polynomials with real non-negative coefficients to be flat a.e.
Spectral ergodic Banach problem and flat polynomials
el Houcein el Abdalaoui
We exhibit a sequence of flat polynomials with coefficients . We thus get that there exist a sequences of Newman polynomials that are -flat, . This settles an…
The Chowla and the Sarnak conjectures from ergodic theory point of view
El Houcein El Abdalaoui, Joanna Kulaga-Przymus, Mariusz Lemanczyk +1
We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences of numbers in {-1,0,1}, and introduce several natural generalizati…
On the pointwise convergence of multiple ergodic averages
E. H. El Abdalaoui
It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space conver…