Density questions in rings of the form 𝒪K[γ] ∩ K

We fix a number field $K$ and study statistical properties of the ring ${{𝒪}}_K[\gamma ]\cap K$ as $\gamma$ varies over algebraic numbers of a fixed degree $n\ge 2$. Given $k\ge 1$, we explicitly compute the density of $\gamma$ for which ${{𝒪}}_K[\gamma ]\cap K={{𝒪}}_K[1/k]$ and show that this does not depend on the number field $K$. In particular, we show that the density of $\gamma$ for which ${{𝒪}}_K[\gamma ]\cap K={{𝒪}}_K$ is $\frac{\zeta (n+1)}{\zeta (n)}$. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define $X(K,\gamma )$ to be a certain finite subset of $\text{Spec}({{𝒪}}_K)$ and show that $X(K,\gamma )$ determines the ring ${{𝒪}}_K[\gamma ]\cap K$. We show that if ${𝔭}_1,{𝔭}_2\in \text{Spec}({{𝒪}}_K)$ satisfy ${𝔭}_1\cap \mathbb{Z}\ne {𝔭}_2\cap \mathbb{Z}$, then the events ${𝔭}_1\in X(K,\gamma )$ and ${𝔭}_2\in X(K,\gamma )$ are independent. As $t\to \infty$, we study the asymptotics of the density of $\gamma$ for which $|X(K,\gamma )|=t$.