Denisty questions in rings of the form ${\mathcal{O}}_K[\gamma ]\cap K$

Abstract

We fix a number field $K$ and study statistical properties of the ring ${\mathcal{O}}_K[\gamma ]\cap K$ 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 ${\mathcal{O}}_K[\gamma ]\cap K={\mathcal{O}}_K[1/k]$ and show that this does not depend on the number field $K$. In particular, we show that the density of $\gamma$ for which ${\mathcal{O}}_K[\gamma ]\cap K={\mathcal{O}}_K$ is $\frac{\zeta (n+1)}{\zeta (n)}$. In a recent paper the authors defined $X(K,\gamma )$ to be a certain finite subset of $\text{Spec}({\mathcal{O}}_K)$ and showed that $X(K,\gamma )$ determines the ring ${\mathcal{O}}_K[\gamma ]\cap K$. We show that if ${\mathfrak{p}}_1,{\mathfrak{p}}_2\in \text{Spec}({\mathcal{O}}_K)$ satisfy ${\mathfrak{p}}_1\cap \mathbb{Z}\ne {\mathfrak{p}}_2\cap \mathbb{Z}$, then the events ${\mathfrak{p}}_1\in X(K,\gamma )$ and ${\mathfrak{p}}_2\in X(K,\gamma )$ are independent. As $t\to \infty$, we study the asymptotics of the density of $\gamma$ for which $|X(K,\gamma )|=t$.