Soient $G$ un groupe abélien compact métrisable, $\Gamma$ son groupe dual et $\Lambda \subset \Gamma$ un ensemble de Rosenthal. Nous montrons que $L_{\Lambda }^{\infty }(G,Y^{*})=C_{\Lambda }(G,Y^{*}),$ lorsque $Y^{*}$ est un espace de Banach ayant la propriété de Radon-Nikodym et $C_{\Lambda }(G,Y^{*})$ est faiblement séquentiellement complet. Nous en déduisons une condition suffisante pour que le produit de deux ensembles de Rosenthal en soit encore un pour le groupe produit. Ensuite nous introduisons la propriété de Radon-Nikodym relative $RN$-$\Lambda$, une généralisation de la propriété de Radon-Nikodym analytique....

A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, $\alpha 1,m,...,m^{d-1}=(m^{d-1}-1)/\left(2(m^d-1)\right)$ and α1,m,m²,... = 1/(2m).

Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets...

Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a “weak ε-Kronecker set” if for every φ:E → there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not “too many”...