In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis,...

This work deals with various questions concerning Fourier multipliers on $L^p$, Schur multipliers on the Schatten class $S^p$ as well as their completely bounded versions when $L^p$ and $S^p$ are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning to the...

Let $E$ be a subset of a discrete abelian group whose compact dual is $G$. $E$ is exactly $p$-Sidon (respectively, exactly non-$p$-Sidon) when$(*)C_E(G{)}^{\wedge }\subset {\ell }^r$ holds if and only if $r\in [p,\infty ]$ (respectively, $r\in (p,\infty )$). $E$ is said to be exactly ${\Lambda }^{\beta }$ (respectively, exactly non-${\Lambda }^{\beta }$) if $E$ has the property${\displaystyle (**)\text{every}f\in L_E^2\left(G\right)\text{satisfies}{\int }_G\exp \left(\lambda \right|f{|}^{2/\alpha }\<\infty ,\text{for}\text{all}\lambda \>0,}$ if and only if $\alpha \in [\beta ,\infty )$ (respectively, $\alpha \in (\beta ,\infty )$).In this paper, for every $p\in [1,2)$ and $\beta \in [1,\infty )$, we display sets which are exactly $p$-Sidon, exactly non-$p$-Sidon, exactly ${\Lambda }^{\beta }$ and exactly non-${\Lambda }^{\beta }$.