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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 $2<p<\infty$. There is well-known that if $S$ is a $\Lambda_p$-set (of W. Rudin) then $$ |S_N|\lesssim_p N^{\frac{2}{p}}.\qquad (1) $$

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 $2<p<\infty$. Recall that a subset $S$ of $\mathbb{Z}$ is a $\Lambda_p$-set if we have $$ \vert\vert f\vert\vert_{L^p(\mathbb{T})}\lesssim \vert\vert f\vert\vert_{L^2(\mathbb{T})} $$ for any trigonometric polynomial $f$ with $\hat{f}(n)=0$ if $n \not\in S$.

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It may help people to answer your question if you give the definition of a $\Lambda_p$ subset of ${\mathbb Z}$ which you know, or which you are using. – user16299 Jun 30 '12 at 22:13

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