The relationship between \(\epsilon\)-Kronecker sets and sidon sets (Q2827405)

Let \(G\) be a compact Abelian group. A subset \(E\) in the dual group \(\Gamma\) is said to be \(\epsilon\)-Kronecker if for every \(\phi : E \to {\mathbb T}\) there exists \(x \in G\) such that \(| \phi (\gamma) - \phi (x)| < \epsilon\) for all \(\gamma \in E\). The Kronecker constant \(\kappa (E)\) of \(E\) is the infimum of such \(\epsilon\).NEWLINENEWLINEThe authors prove that if \(\kappa (E) < 2\), then \(E\) is a Sidon set.NEWLINENEWLINEFor that they use Pisier's entropy characterization of Sidon sets [\textit{G. Pisier}, in: Topics in modern harmonic analysis, Proc. Semin., Torino and Milano 1982, Vol. II, 911--944 (1983; Zbl 0539.43004)]: \(E\) is a Sidon set if and only if there is \(\epsilon > 0\) such that, for every finite subset \(F \subseteq E\), there is \(Y \subseteq G\) with \(|Y| \geq 2^{\epsilon |F|}\) such that \(\epsilon \leq \sup_{\gamma \in F} |\gamma (x) - \gamma (y)|\) whenever \(x \neq y \in Y\) (see [\textit{D. Li} and \textit{H. Queffélec}, Introduction à l'étude des espaces de Banach. Analyse et Probabilités. Cours Spécialisés 12, Société Mathématique de France (2004), Chapitre 13, Théorème V.5]).NEWLINENEWLINEThey also give examples of Sidon sets \(E\) with \(\kappa (E) = 2\).