Continuous measures and lacunarity on hypergroups

The relationship between measures on a compact hypergroup $K$ whose Fourier-Stieltjes transforms vanish at infinity and the space ${M_c}(K)$ of continuous measures is studied. Examples are provided of measures $\mu$ with $\hat \mu$ vanishing at infinity and $\mu \in {M_c}(K)$. Sufficient conditions are given for $\hat \mu \in {c_0}(\hat K)$ to imply $\mu \in {M_c}(K)$. An investigation of Helson sets on compact abelian hypergroups is initiated and the study of Sidon sets on compact abelian hypergroups is continued. A class of compact abelian hypergroups is shown to have no infinite Helson sets and no infinite Sidon sets. This result generalizes results of D. L. Ragozin and D. Rider on central Sidon sets for compact connected Lie groups.