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The product of two Dirichlet series

Frédéric Bayart — 2004

Acta Arithmetica

Topological and algebraic genericity of divergence and universality

Frédéric Bayart — 2005

Studia Mathematica

We give general theorems which assert that divergence and universality of certain limiting processes are generic properties. We also define the notion of algebraic genericity, and prove that these properties are algebraically generic as well. We show that universality can occur with Dirichlet series. Finally, we give a criterion for the set of common hypercyclic vectors of a family of operators to be algebraically generic.

Boundary behavior and Cesàro means of universal Taylor series.

Frédéric Bayart — 2006

Revista Matemática Complutense

We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its Cesàro means are universal.

The multifractal box dimensions of typical measures

Frédéric Bayart — 2012

Fundamenta Mathematicae

We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^{d}$ . Our results are new even in the context of box dimensions of measures.

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart; Yanick Heurteaux — 2012

Annales scientifiques de l'École Normale Supérieure

A famous theorem of Carleson says that, given any function $f \in L^{p} (𝕋)$ , $p \in (1, + \infty)$ , its Fourier series $(S_{n} f (x))$ converges for almost every $x \in 𝕋$ . Beside this property, the series may diverge at some point, without exceeding $O (n^{1 / p})$ . We define the divergence index at $x$ as the infimum of the positive real numbers $β$ such that $S_{n} f (x) = O (n^{β})$ and we are interested in the size of the exceptional sets $E_{β}$ , namely the sets of $x \in 𝕋$ with divergence index equal to $β$ . We show that quasi-all functions in $L^{p} (𝕋)$ have a multifractal behavior with respect to this definition....

Sur l'irréductibilité dans l'anneau des séries de Dirichlet analytiques.

Frédéric Bayart; Augustin Mouze — 2005

Publicacions Matemàtiques

We discuss some local analytic properties of the ring of Dirichlet series. We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one. In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series. Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.

(Non-)weakly mixing operators and hypercyclicity sets

Frédéric Bayart; Étienne Matheron — 2009

Annales de l’institut Fourier

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space $ℓ^{1} (ℕ)$ , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_{0} (ℕ)$ or $ℓ^{p} (ℕ)$ , $1 < p < \infty$ . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

Division et composition dans l'anneau des séries de Dirichlet analytiques

Frédéric Bayart; Augustin Mouze — 2003

Annales de l'Institut Fourier

Ce travail est une étude analytique locale de l’anneau des séries de Dirichlet convergentes. Dans un premier temps, on établit des propriétés arithmétiques de cet anneau ; on prouve en particulier sa factorialité, que l’on déduit de théorèmes de division du type Weierstrass. Ensuite, on s’intéresse à des problèmes de composition. Soient $f (s)$ et $ϕ (s)$ des séries de Dirichlet convergentes. On sait que $f (c_{0} s + ϕ (s)),$ avec $c_{0} \in ℕ^{*},$ est encore une série de Dirichlet convergente. On étudie la réciproque : sous les hypothèses que...