Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, meaning its natural density in $\mathbb{Z}$ is zero. Furthermore, suppose there exist congruence conditions modulo $m$ that all elements of $O_a$ satisfy. Let $p$ be a prime greater than 3, and let $u$ be a fixed positive integer coprime to $m$. Define the family $S_{p,u} = \{un^p : n \in \mathbb{Z}\}$.
We are interested in the possibility of $S_{p,u}$ forming a higher-order obstruction to $O_a$. Specifically, we want to determine if there exists a group $G$, an action of $G$ on $\mathbb{Z}$, an initial integer $a$, a prime $p > 3$, and an integer $u$ such that the following two conditions hold:
Infinitely many elements of $S_{p,u}$ satisfy the congruence conditions modulo $m$ associated with $O_a$.
No element of $S_{p,u}$ appears in the orbit $O_a$.
Can we define a non-trivial group homomorphism $\chi_p: G \to \mu_p$, where $\mu_p$ is the group of $p$-th roots of unity, such that for all $g \in G$ and all $n \in \mathbb{Z}$ for which $un^p$ is admissible modulo $m$, we have:
$$\chi_p(g) \neq \left(\frac{g \cdot a}{un^p}\right)_p$$
where $(\frac{\cdot}{\cdot})_p$ denotes the $p$-th power residue symbol in a suitably chosen number field containing the $p$-th roots of unity? Such a homomorphism could be considered a generalized "obstruction symbol" detecting the absence of $p$-th power families in the orbit.
