Sidon sets with extremal Sidon constants

Abstract:

A finitely supported measure $\mu$ on an l.c.a. group is said to be extremal if ${\left \| {\hat \mu } \right \|_\infty } = {\left \| \mu \right \|^{1/2}} = {(\# {\text {supp}}\;\mu {\text {)}}^{1/2}}$. If $\mu$ is an extremal measure and $E$ is the support of $\mu$, it follows that the Sidon constant of $E$ is ${(\# E)^{1/2}}$, in which case $E$ is also said to be extremal. Our results are these. (1) An "independent" union of $m$ cosets of a finite subgroup $H$ of $G$ is extremal if and only if (essentially) $m$ divides $\# H$. (2) Not all extremal subsets of abelian groups have the form described in (1). (3) For any group (abelian or not), the Sidon constant of that group is at least $(.8){(\# G)^{1/13}}$.