An invariant mean \(M\) is a positive linear functional on \(L^\infty ({\mathbb T})\) such that \(M (1) = 1\) and which is invariant by translation: \(M (f_t) = M (f)\) for every \(f \in L^\infty ({\mathbb T})\) and every \(t \in {\mathbb T}\). One says that \(f \in L^\infty ({\mathbb T})\) has a unique invariant mean if \(M (f) = \int_{\mathbb T} f (t)\, dt\) for every invariant mean. It is clear that every continuous function, as well as every Riemann-integrable function, has a unique invariant mean, but M. Talagrand [Proc. Am. Math. Soc. 82, 253–256 (1981; Zbl 0472.28007)] proved that there are non Riemann-integrable functions with a unique invariant mean. One says that a set \(\Lambda \subseteq {\mathbb Z}\) is an \(LP\)-set if for every \(f \in L^\infty ({\mathbb T})\) whose spectrum is contained in \(\Lambda\) (in short \(f \in L^\infty_\Lambda\)) and every \(n \in {\mathbb Z}\) the function \(\text{e}^{2\pi i n t} f (t)\) has a unique invariant mean. Every Rosenthal set \(\Lambda\) (a set for which every \(f \in L^\infty_\Lambda\) is continuous) is an \(LP\)-set and F. Lust-Piquard [Colloq. Math. 58, No. 1, 29–38 (1989; Zbl 0694.43005)] showed that the intersection of the set \({\mathbb P}\) of the primes with the arithmetical progression \(5{\mathbb Z} + 2\) is an \(LP\)-set which is not a Rosenthal set. Her proof used that \(LP\)-sets are localisable: \(\Lambda\) is an \(LP\)-set whenever, for every \(n \in {\mathbb Z}\), there is a neighbourhood \(V_n\) of \(n\) for the Bohr topology such that \(\Lambda \cap V_n\) is an \(LP\)-set. She had to cut \({\mathbb P}\) with \(5{\mathbb Z} + 2\) to avoid \(-1\) and \(+1\) which are cluster points of \({\mathbb P}\) [see S. Hartman, Colloq. Math. 42, 209–222 (1979; Zbl 0432.43005)]. It should be noticed that \({\mathbb N}\) is not an \(LP\)-set, as shown by Y. Katznelson in 1975 (this also follows from a previous result of the authors [J. Funct. Anal. 233, No. 2, 545–560 (2006; Zbl 1089.43003)]).
In the paper under review, the authors show that a weaker notion of localisation ensures that \(\Lambda\) be an \(LP\)-set; that allows us to show that the whole \({\mathbb P}\) is \(LP\), and moreover that, for every \(r \geq 1\), the set \(F_r\) of the integers whose decomposition in prime numbers has at most \(r\) factors is an \(LP\)-set. They show that the \(F_r\)’s are actually strong \(LP\)-sets: their closure is an \(LP\)-set yet (actually, they prove that \(F_r\) is closed). They show that the union of an \(LP\)-set with a strong \(LP\)-set is again an \(LP\)-set. For \(\Lambda \subseteq {\mathbb N}\), they show that an \(LP\)-set cannot be uniformly distributed (\(\Lambda =\{ 0 \leq \lambda_0 < \lambda_1 < \cdots \}\) is uniformly distributed if \({1 \over N+1} \sum_{j=0}^N \text{e}^{2\pi i {\lambda_j} t} @>>N \to \infty> 0\) for every \(t \neq 0\) (note that \({\mathbb P}\) is not uniformly distributed, in spite of Vinogradov’s theorem)) and, using the main result of D. Li, H. Queffélec and L. Rodríguez-Piazza [J. Anal. Math. 86, 105–138 (2002; Zbl 1018.43004)], that there exist sets which are almost surely \(p\)-Sidon for every \(p > 1\), of uniform convergence, \(\Lambda (q)\) for every \(q < \infty\), uniformly distributed, but not \(LP\).
In the last section, they study sets \(\Lambda\) for which \({\mathcal C}_\Lambda ({\mathbb T})\) has Pełczyński’s property \((V)\). They show in particular that this cannot happen if \(\Lambda\) is a \(p\)-Sidon set with \(1 \leq p < 4/3\) (because such sets are a finite union of sets whose pace tends to infinity: this is due to M. Déchamps-Gondim [Ann. Inst. Fourier 22, No. 3, 51–79 (1972; Zbl 0273.43010)] for \(p = 1\), but her proof works for \(1 \leq p < 4/3\)).
The paper ends with several questions.