Nontriviality of rings of integral-valued polynomials

Let $S$ be a subset of $\bar{\mathbb{Z}}$, the ring of all algebraic integers. A polynomial $f\in \mathbb{Q}[X]$ is said to be integral-valued on $S$ if $f(s)\in \bar{\mathbb{Z}}$ for all $s\in S$. The set ${\text{Int}}_{\mathbb{Q}}(S,\bar{\mathbb{Z}})$ of all integral-valued polynomials on $S$ forms a subring of $\mathbb{Q}[X]$ containing $\mathbb{Z}[X]$. We say that ${\text{Int}}_{\mathbb{Q}}(S,\bar{\mathbb{Z}})$ is trivial if ${\text{Int}}_{\mathbb{Q}}(S,\bar{\mathbb{Z}})=\mathbb{Z}[X]$, and nontrivial otherwise. We give a collection of necessary and sufficient conditions on $S$ in order ${\text{Int}}_{\mathbb{Q}}(S,\bar{\mathbb{Z}})$ to be nontrivial. Our characterizations involve, variously, topological conditions on $S$ with respect to fixed extensions of the $p$-adic valuations to $\bar{\mathbb{Q}}$; pseudo-monotone sequences contained in $S$; ramification indices and residue field degrees; and the polynomial closure of $S$ in $\bar{\mathbb{Z}}$.