For each $i \in \mathbb{N}$, let $G_{i}$ be a finite abelian group and $\widehat{G_{i}}$ the $\overline{\mathbb{Q}}$-valued character group of $G_{i}$. Suppose that $|G_{i}| \rightarrow \infty$ as $i \rightarrow \infty$. Let $S_{i} \subset \widehat{G_{i}}$ be a subset of characters satisfying the following.
(i) For given $\epsilon > 0$, we have $\langle s | s \in S_{i} \rangle \gg |G_{i}|^{1-\epsilon}$ as $i \rightarrow \infty$. (ii) $S_{i}$ is $G_{\mathbb{Q}}$-stable i.e. if $s\in S_{i}$, then $s^{\sigma} \in S_{i}$ for each $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with $s^{\sigma}$ being the conjugate character. (iii) There exists a finite set $T$ independent of $(G_{i})_{i=1}^{\infty}$ and non-zero scalars $a_{i,j} \in \overline{\mathbb{Q}}$ dependent on $G_{i}$ such that $$Im(\sum_{j=1}^{|S|}a_{i,j}s_{j}) \subset T.$$ Here we view the linear combination $\sum_{j=1}^{|S|}a_{i,j}s_{j}$ a function on $G_{i}$.
Can we obtain an estimate for a lower bound on $|S|$? Is it reasonable to hope for a bound of the form $|G|^{\delta}$ for some $\delta > 0$?
