We denote by $|A|$ the cardinal of a set $A$. Let $S$ be a subset of $\mathbb{Z}$. Denote $S_N=S\cap [0,N]$ where $N$ is an integer. Suppose $2
1) Let $G$ be a compact abelian group. Does there exist generelizations of this property for $\Lambda_p$-sets for the dual group $\Gamma$ of $G$?
2) I know that J. Bourgain used the property (1) in order to show that there exists $\Lambda_p$-set which are not $\Lambda_q$-set for any $q>p$. In a survey (Handbook of Banach spaces), Bourgain says that his theorem is also true if $\mathbb{T}$ is replaced by any infinite compacte abelian group $G$.
What is the substitute of (1) used by Bourgain for deal with compact abelian groups?
Add-on:Suppose $2