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el Houcein el Abdalaoui

Publications and source records attributed to el Houcein el Abdalaoui.

At least 19 recordsLinked to original sources
arXiv2026

On the spectral analysis of dynamical Möbius-Sarnak process and topological entropy of Möbius fonction

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 countable Lebesgue component. Inspired by recent work of M. Lin and the author, we extend the notion of spectral measure to all operators on Banach spaces. This generalization is further motivated by the Bellow-Losert extension of Wiener's notion of the spectral measure of sequences. Furthermore, we establish unconditionally that the topological entropy of the Möbius flow is given by $\frac{6}{π^2}\log 3$. Among other consequences, we recover a recent result by el Abdalaoui-Nerurkar which asserts that for any quasi-generic measure for the Möbius function, the Möbius flow equipped with this measure has a countable Lebesgue component in its spectrum. It follows that the Sarnak Möbius orthogonality conjecture holds for any topological dynamical system with singular spectrum. We further show that all the potential spectral measures of he Möbius function are absolutely continuous with respect to Lebesgue measure.

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arXiv2026

Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

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 accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of `affinity between measures', (or the so-called `Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{ä}ki-Radziwiłl-Tao's theorem and Matom{ä}ki-Radziwiłl's theorem on the correlations of order two of the Liouville function.

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arXiv2025

A generalization of Littlewood's $L^α$ flat theorem, $α>0$

We establish a generalization of Littlewood's criterion on $L^α$-flatness by proving that there is no $L^α$-flat polynomials, $α>0$, within the class of analytic polynomials on the unit circle of the form $ P_n(z)=\sum_{m=1}^{n}c_m z^m, n \in {\mathbb{N}}^*,$ satisfying $$ \sum_{m=1}^{n}|c_m|^2 \leq \frac{K}{n^2} \sum_{m=1}^{n}m^2 |c_m|^2, $$ where $K$ is an absolutely constant. As a consequence, we confirm the $L^α$-Littlewood conjecture, and thereby the $L^1$-Newman and $L^\infty$-Erdös conjectures. Our approach combines the $L^α$ Littlewood theorem with the generalized Clarkson's second inequality for $L^α(X,\mathcal{A},m;B)$, with $B$ a Banach spaces and $1 < α\leq 2.$ It follows that there are only finitely many Barker sequences, and we further present several applications in number theory and the spectral theory of dynamical systems. Finally, we construct Gauss-Fresnel polynomials that are Mahler-flat, providing a new proof of the Beller-Newman theorem.

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arXiv2025

On $L^α$-flatness of Erdős-Littlewood's polynomials

It is shown that Erdös--Littlewood's polynomials are not $L^α$-flat when $α> 2$ is an even integer (and hence for any $α\geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erdös--Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erdös--Littlewood polynomials. Our proof is short and simple. It relies on the classical lemma for $L^p$ norms of the Dirichlet kernel, the Marcinkiewicz--Zygmund interpolation inequalities, and the $p$-concentration theorem due to A. Bonami and S. Révész.

math.CA↗
arXiv2025

On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights

We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbolν$, for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, for any integers $a,b$, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have \[\frac{1}{N} \sum_{n=1}^{N} \boldsymbolν(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.\] We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.

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arXiv2023

Mean ergodic theorems in $L^r(μ)$ and $H^r(\mathbb T)$, $0<r<1$

Let $T$ be the Koopman operator of a measure preserving transformation $θ$ of a probability space $(X,Σ,μ)$. We study the convergence properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in L^r(μ)$, $0<r<1$. We prove that if $\int |M_nf|^r dμ\to 0$, then $f \in \overline{(I-T)L^r}$, and show that the converse fails whenever $θ$ is ergodic aperiodic. When $θ$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\int |M_nf|^r dμ\to 0$). We further establish that for $1 \leq p <\frac{1}{r},$ there is a dense $G_δ$ subset ${\mathcal F}\subset L^p(X,μ)$ such that $\limsup_n \frac{|T^nh|}{n^r}=\infty$ a.e. for any $h \in {\mathcal F}$.

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arXiv2023

Spectral ergodic Banach problem and flat polynomials

We exhibit a sequence of flat polynomials with coefficients $0,1$. We thus get that there exist a sequences of Newman polynomials that are $L^α$-flat, $0 \leq α<2$. This settles an old question of Littlewood. In the opposite direction, we prove that the Newman polynomials are not $L^α$-flat, for $α\geq 4$. We further establish that there is a conservative, ergodic, $σ$-finite measure preserving transformation with simple Lebesgue spectrum. This answer affirmatively a long-standing problem of Banach from the Scottish book. Consequently, we obtain a positive answer to Mahler's problem in the class of Newman polynomials, and this allows us also to answer a question raised by Bourgain on the supremum of the $L^1$-norm of $L^2$-normalized idempotent polynomials.

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arXiv2023

$L^1$-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof

We present a simple proof on the existence of $L^1$-flat analytic polynomials with coefficients $0,1$ on the circle and on the real line and we give an example of a conservative ergodic map and flow whose unitary operators admits a simple Lebesgue spectrum. Among other results, we obtain an answer to Bourgain's question on the supremum of $L^1$-norm of such polynomials and to a question inspired by Lehmer's problem on the supremum of the Mahler measures of those polynomials.

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arXiv2022

On Veech's proof of Sarnak's theorem on the Möbius flow

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 conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the logarithmic Sarnak conjecture is equivalent to logaritmic Chowla conjecture, furthermore, if the even logarithmic Sarnak's conjecture is true then there is a subsequence with logarithmic density one along which Chowla conjecture holds, that is, the Möbius function is quasi-generic.

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arXiv2022

Mixing sequences for non-mixing transformations and group actions

We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences $(s_i)$ which satisfy the Rajchman dissociated property. Our examples are based on the staircase rank one construction, $M$-towers constructions and the Gaussian transformations. As a consequence, we obtain there are non-mixing maps which are mixing along the squares. We further prove that a sequence $M=(m_n)$ is a mixing sequence for some weak mixing ${1}/{2}$-rigid transformation $T$ if and only if the complement of $M$ is a thick set. This result is generalized to ${r}/{(r+1)}$-rigid transformations for $r\in \mathbb{N}$. Moreover, by applying Host-Parreau characterization of the set of continuity from Harmonic Analysis, we extend our results to the infinite countable abelian group actions.

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arXiv2022

Exponential sums of the Möbius function and flat polynomials

We present an elementary problem on analytic polynomials with coefficients $\pm 1$ or in $\{0,\pm 1 \}$ which implies Riemann hypothesis. It is turns out that this problem is a particular case of the weak form of a flat polynomials problem asked by Erdös in his 1957's paper. We provide also under an extra-condition the converse.

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arXiv2021

On the Mahler measure of the spectrum of rank one maps

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 only if its Mahler measure is zero. As a consequence, we exhibit a new subclass of rank one maps acting on a finite measure space with singular spectrum. In our proof the $H^p$ theory coming to play. Furthermore, by appealing to a deep result of Bourgain, we prove that the Mahler measure of the spectrum of rank one map with cutting parameter $p_n=O(n^β)$, $β\leq 1$ is zero, and we establish that the integral of the absolute part of any generalized Riesz-product is strictly less than 1. This answer partially a question asked by M. Nadkarni.

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arXiv2021

A disproof of $L^α$ polynomials Rudin conjecture, $2 \leq α<4.$

It is shown that the $L^α$-norms polynomials Rudin conjecture fails. Our counterexample is inspired by Bourgain's work on NLS. Precisely, his study of the Strichartz's inequality of the $L^6$-norm of the periodic solutions given by the two dimension Weyl sums. We gives also a lower bound of the $L^α$-norm of such solutions for $α\neq 2$. As a consequence, we establish that for any $0<a<b,$ the following set $E(a,b)=\Big\{(x,t) \in \mathbb{T}^2 \; : \;a \sqrt{N} \leq \Big|\sum_{n=1}^{N}e(n^2t+nx)\Big| \leq b \sqrt{N} \; \textrm{~infinitely~often}\;\Big\},$ has a Lebesgue measure $0$. We further present an alternative proof of Cordoba's theorem based on Paley-Littlewood inequalities.

math.CA↗
arXiv2021

Van der Corput inequality for real line and Wiener-Wintner theorem for amenable groups

We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $\mathbb{R}$-action which assert that for any family of maps $(T_t)_{t \in \mathbb{R}}$ acting on the Lebesgue measure space $(Ω,{\cal {A}},μ)$ where $μ$ is a probability measure and for any $t\in \mathbb{R}$, $T_t$ is measure-preserving transformation on measure space $(Ω,{\cal {A}},μ)$ with $T_t \circ T_s =T_{t+s}$, for any $t,s\in \mathbb{R}$. Then, for any $f \in L^1(μ)$, there is a a single null set off which $\displaystyle \lim_{T \rightarrow +\infty} \frac1{T}\int_{0}^{T} f(T_tω) e^{2 i πθt} dt$ exists for all $θ\in \mathbb{R}$. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Weiss and Ornstein.

math.DS↗
arXiv2021

On the symmetric version of Seaki Theorem and flat densities

It is shown that for any $α\in ]\frac12,1[$ there exists a symmetric probability measure $σ$ on the torus such that the Hausdorff dimension of the support of $σ$ is $α$ and $σ*σ$ is absolutely continuous with flat continuous Radon-Nikodym derivative. Namely, we obtain a symmetric version of Seaki Theorem but the flat Radon-Nikodym derivative of $σ*σ$ can not be a Lipschitz function.

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arXiv2020

On the spectral type of rank one flows and Banach problem with calculus of generalized Riesz products on the real line

It is shown that a certain class of Riesz product type measures on $\mathbb{R}$ is realized a spectral type of rank one flows. As a consequence, we will establish that some class of rank one flows has a singular spectrum. Some of the results presented here are even new for the $\mathbb{Z}$-action. Our method is based, on one hand, on the extension of Bourgain-Klemes-Reinhold-Peyrière method, and on the other hand, on the extension of the Central Limit Theorem approach to the real line which gives a new extension of Salem-Zygmund Central Limit Theorem. We extended also a formula for Radon-Nikodym derivative between two generalized Riesz products obtained by el Abdalaoui-Nadkarni and a formula of Mahler measure of the spectral type of rank one flow but in the weak form. We further present an affirmative answer to the flow version of the Banach problem, and we discuss some issues related to flat trigonometric polynomials on the real line in connection with the famous Banach-Rhoklin problem in the spectral theory of dynamical systems.

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arXiv2020

On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have $$\lim_{\bar{N} \longrightarrow +\infty} \frac{1}{N} \sum_{n=1}^{N}\boldsymbolν(n) f(T^{P(n)}x)g(T^{Q(n)}x)=0$$ where $\boldsymbolν$ is the Liouville function or the Möbius¶ function.

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