Let $G$ be a finite abelian group and $\widehat{G}$ the character group. Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ for any $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $H_{S} \subset \widehat{G}$ be the subgroup generated by $S$.

We now consider a sequence $(G_{i},S_{i})$ as above with $|G_{i}|\rightarrow \infty$. Suppose that there exists $\epsilon$ with $ 0 < \epsilon < 1$ such that $|H_{S_{i}}| \gg |G_{i}|^{\epsilon}$. Then, is it necessary to have $|S_{i}| \gg \ln(|G_{i}|)$? Can we say something about optimal lower bound?