Extreme values for the Sidon constant

Abstract:

Let $G$ be a compact group and let $\phi \ne P \subseteq \hat G$. We consider the inequalities $1 \leqslant \kappa (P) \leqslant {({\Sigma _{\sigma \in P}}d_\sigma ^2)^{1/2}}$, where $\kappa (P)$ denotes the Sidon constant of $P$. The condition $\kappa (P) = 1$ essentially characterizes an example of Figà-Talamanca and Rider. The condition $\kappa (P) = {({\Sigma _{\sigma \in P}}d_\sigma ^2)^{1/2}}$ for finite $P$ is equivalent to the existence of certain interesting functions on $G$. We show that $\kappa (\hat G) = {\left | G \right |^{1/2}}$ for a very large class of finite groups $G$, and this implies the existence of "$G$-circulant" unitary matrices whose entries all have modulus ${\left | G \right |^{ - 1/2}}$.