Soit $D$ un domaine plan ; y a-t-il des suites $z_n$ telles que toute suite bornée $w$ puisse être interpolée en $z_n$ par une fonction $f\left(z\right)$ régulière et bornée dans $D$ ? Dans l’affirmative est-il vrai que toute suite $z_n$ qui tend assez rapidement vers la frontière de $D$ possède cette propriété ? On répond affirmativement à ces deux questions dans le cas où $D$ est le cercle-unité.

Let $f\left(x\right)$ be a power series ${\sum }_{n\ge 1}\zeta \left(n\right)x^{e\left(n\right)}$, where $\left(e\right(n\left)\right)$ is a strictly increasing linear recurrence sequence of non-negative integers, and $\left(\zeta \right(n\left)\right)$ a sequence of roots of unity in ${\overline{\mathbb{Q}}}_p$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ${\mathbb{Q}}_p$ of the elements $f\left({\alpha }_1\right),...,$ $f\left({\alpha }_t\right)$ from ${ℂ}_p$ in terms of the distinct ${\alpha }_1,...,{\alpha }_t\in {\mathbb{Q}}_p$ satisfying $0\<|{\alpha }_{\tau }{|}_p\<1$ for $\tau =1,...,t$. A striking application of our basic result says that, in the case $e\left(n\right)=n$, the set $\left\{f\left(\alpha \right)\right|\alpha \in {\mathbb{Q}}_p,0\<|\alpha {|}_p\<1\}$ is algebraically independent over ${\mathbb{Q}}_p$ if...

Let $ℬ\left(\mathscr{H}\right)$ be the set of all bounded linear operators acting in Hilbert space $\mathscr{H}$ and ${ℬ}^{+}\left(\mathscr{H}\right)$ the set of all positive selfadjoint elements of $ℬ\left(\mathscr{H}\right)$. The aim of this paper is to prove that for every finite sequence ${\left(A_i\right)}_{i=1}^n$ of selfadjoint, commuting elements of ${ℬ}^{+}\left(\mathscr{H}\right)$ and every natural number $p\ge 1$, the inequality $$\frac{e^p}{p^p}\left(\sum _{i=1}^n{A}_i^p\right)\le exp\left(\sum _{i=1}^n{A}_i\right),$$ holds.

Let $T:x\mapsto x+g$ be an ergodic translation on the compact group $C$ and $M\subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mathbf{a}:\mathbb{Z}\mapsto \{0,1\}$ defined by $\mathbf{a}\left(k\right)=1$ if $T^k\left(0_C\right)\in M$ and $\mathbf{a}\left(k\right)=0$ otherwise, is called a Hartman sequence. This paper studies the growth rate of $P_{\mathbf{a}}\left(n\right)$, where $P_{\mathbf{a}}\left(n\right)$ denotes the number of binary words of length $n\in \mathbb{N}$ occurring in $\mathbf{a}$. The growth rate is always subexponential and this result is optimal. If $T$ is an ergodic translation $x\mapsto x+\alpha$ $(\alpha =({\alpha }_1,...,{\alpha }_s))$ on ${𝕋}^s$ and $M$ is a box with...