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Let $E$ be a subset of a discrete abelian group whose compact dual is $G$. $E$ is exactly $p$-Sidon (respectively, exactly non-$p$-Sidon) when$(*)C_E(G{)}^{\wedge }\subset {\ell }^r$ holds if and only if $r\in [p,\infty ]$ (respectively, $r\in (p,\infty )$). $E$ is said to be exactly ${\Lambda }^{\beta }$ (respectively, exactly non-${\Lambda }^{\beta }$) if $E$ has the property${\displaystyle (**)\text{every}f\in L_E^2\left(G\right)\text{satisfies}{\int }_G\exp \left(\lambda \right|f{|}^{2/\alpha }\<\infty ,\text{for}\text{all}\lambda \>0,}$ if and only if $\alpha \in [\beta ,\infty )$ (respectively, $\alpha \in (\beta ,\infty )$). In this paper, for every $p\in [1,2)$ and $\beta \in [1,\infty )$, we display sets which are exactly $p$-Sidon, exactly non-$p$-Sidon, exactly ${\Lambda }^{\beta }$ and exactly non-${\Lambda }^{\beta }$.