Papers by houcein el Abdalaoui
arXiv (Cornell University), Dec 30, 2023
Let T be the Koopman operator of a measure preserving transformation θ of a probability space (X,... more 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 n f := 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 ∈ (I -T )L r for which M n f r does not converge a.e. (although |M n f | r dµ → 0). We further establish that for 1 ≤ p < 1 r , there is a dense When T is induced by an irrational rotation of T, the Hardy spaces H r (T) are T -invariant. For 0 < r < 1, we prove that H r (T) = {constants} ⊕ (I -T )H r (T), and |M n f | r dµ → 0 for f ∈ (I -T )H r (T). However, there exists f ∈ (I -T )H r (T) such that M n f does not converge a.e.
arXiv (Cornell University), Oct 23, 2019
It is shown that if the analytic polynomials with Möbius are L p-semi-flat then the Riemann hypot... more It is shown that if the analytic polynomials with Möbius are L p-semi-flat then the Riemann hypothesis holds. It turns out that this problem is a particular case of the weak form of flat polynomials problem asked by Erdös in his 1957's paper. Under an extra-condition, we establish also the converse. We further point out an oversight in Littlewood paper [14] and we correct it. It is also shown that the recent flat polynomials in Littlewood sense constructed by P. Balister and al. are not L α-flat, for any α ≥ 0.
Chowla and Sarnak conjectures for Kloosterman sums
Mathematische Nachrichten
We formulate several analogs of the Chowla and Sarnak conjectures, which are widely known in the ... more We formulate several analogs of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.
We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourga... more 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, B, µ), for any f ∈ L r (X, µ) , g ∈ L r ′ (X, µ) such that 1/r + 1/r′ = 1, for any nonconstant polynomials P(n), Q(n), n ∈ Z, taking integer values, and for almost all x ∈ X, we have, 1/N ΣN n=1 f(T^P(n) x) and 1/(πN) Σp≤N p prime f(T^P (p)x)g(T^Q(p)x), converge. Here πN is the number of prime in [1, N]
HAL (Le Centre pour la Communication Scientifique Directe), Jul 7, 2020
It is shown that a certain class of Riesz product type measures on R is realized a spectral type ... more It is shown that a certain class of Riesz product type measures on 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 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.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 29, 2021
We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-produc... more 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 pn = O(n β), β ≤ 1 is zero, and we establish that the integral of the square root of the absolute part of any generalized Riesz-product is strictly less than 1. This answer partially a question asked by M. Nadkarni. What matters to an active man is to do the right thing; whether the right thing comes to pass should not bother him. Goethe Fejér used to say-in the 1930's, "Everybody writes and nobody reads." This was true eventhen. Reviewing has improved, but even so it is very hard.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 26, 2020
It is shown that for any α ∈] 1 2 , 1[ there exists a symmetric probability measure σ on the toru... more It is shown that for any α ∈] 1 2 , 1[ there exists a symmetric probability measure σ on the torus such that the Hausdorff dimension of its support 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.
Uniform distribution theory, 2020
We exhibit a class of Littlewood polynomials that are not L α -flat for any α ≥ 0. Indeed, it is ... more We exhibit a class of Littlewood polynomials that are not L α -flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not L α -flat, α ≥ 0, when the frequency of −1 is not in the interval ] 1 4 {1 \over 4} , 3 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 L α -flat for any α> 2 if the frequency of −1 is not 1 2 {1 \over 2} . Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not L α -flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.
Constructive Mathematical Analysis, 2021
We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a... more 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 RR-action which assert that for any family of maps (Tt)t∈R(Tt)t∈R acting on the Lebesgue measure space (Ω,A,μ)(Ω,A,μ), where μμ is a probability measure and for any t∈Rt∈R, TtTt is measure-preserving transformation on measure space (Ω,A,μ)(Ω,A,μ) with Tt∘Ts=Tt+sTt∘Ts=Tt+s, for any t,s∈Rt,s∈R. Then, for any f∈L1(μ)f∈L1(μ), there is a single null set off which $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$limT→+∞1T∫0Tf(Ttω)e2iπθtdt exists for all θ∈θ∈\RRR. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.
arXiv: Dynamical Systems, 2015
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Papers by houcein el Abdalaoui