Let $G = \mathbb Z/n\mathbb Z$ and let $\emptyset\neq S\subseteq G$. Then the Cayley digraph of $G$ with respect to $S$ has vertex set $G$ and directed edges of the form $g\rightarrow gs$ with $s\in S$. I don't assume $S$ is symmetric or that $S$ generates $G$. Let $A_S$ be the adjacency matrix of the corresponding Cayley digraph. Let us say $S$ is good if $A_S$ is invertible.

Define $p_S(x)=\sum_{k\in S}x^k$ where I identify $G$ with $\lbrace 0,\ldots, n-1\rbrace$ in the usual way. Discrete Fourier analysis says $S$ is good iff $p_S$ has no root which is an $n^{th}$-root of unity, i.e., $p_S$ is relatively prime to $x^n-1$. Alternatively, $S$ is good if and only if $\sum_{s\in S}s$ is an invertible element of the group ring $\mathbb CG= \mathbb C[x]/(x^n-1)$.

Obviously if $n$ is prime, then all proper non-empty subsets are good since the cyclotomic polynomial is $p_G$. On the other hand, if $S$ is a proper subgroup of $G$, then it is easy to see that $S$ is not good. Let us say that for $S,T\subseteq G$ the sum $S+T$ is unambiguous if each element of $S+T$ can uniquely be expressed as a sum of an element of $S$ with an element of $T$. For example, a coset $g+H$ is unambiguous. Clearly, if $S+T$ is unambiguous, then $p_{S+T}=p_Sp_T\bmod (x^n-1)$ and so if either $S$ or $T$ is bad, then so is $S+T$.

Question. Is there some nice characterization of all good subsets of $G$ as, say, built from proper subgroups via unambiguous sums and perhaps some other operations?

Let $G = \mathbb Z/n\mathbb Z$ and let $\emptyset\neq S\subseteq G$. Then the Cayley digraph of $G$ with respect to $S$ has vertex set $G$ and directed edges of the form $g\rightarrow gs$ with $s\in S$. I don't assume $S$ is symmetric or that $S$ generates $G$. Let $A_S$ be the adjacency matrix of the corresponding Cayley digraph. Let us say $S$ is good if $A_S$ is invertible.

Define $p_S(x)=\sum_{k\in S}x^k$ where I identify $G$ with $\lbrace 0,\ldots, n-1\rbrace$ in the usual way. Discrete Fourier analysis says $S$ is good iff $p_S$ has no root which is an $n^{th}$-root of unity, i.e., $p_S$ is relatively prime to $x^n-1$. Alternatively, $S$ is good if and only if $\sum_{s\in S}s$ is an invertible element of the group ring $\mathbb CG= \mathbb C[x]/(x^n-1)$.

Obviously if $n$ is prime, then all proper non-empty subsets are good since the cyclotomic polynomial is $p_G$. On the other hand, if $S$ is a proper subgroup of $G$, then it is easy to see that $S$ is not good. Let us say that for $S,T\subseteq G$ the sum $S+T$ is unambiguous if each element of $S+T$ can uniquely be expressed as a sum of an element of $S$ with an element of $T$. For example, a coset $g+H$ is unambiguous. Clearly, if $S+T$ is unambiguous, then $p_{S+T}=p_Sp_T\bmod (x^n-1)$ and so if either $S$ or $T$ is bad, then so is $S+T$.

Question. Is there some nice characterization of bad subsets of $G$ as, say, built from proper subgroups via unambiguous sums and perhaps some other operations?