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

TL;DR

This paper investigates the statistical distribution of algebraic numbers in number fields that generate specific rings, computing densities of certain ring structures and analyzing the independence of prime-related events.

Contribution

It explicitly computes densities of algebraic numbers generating particular rings in number fields and establishes independence properties of prime-related events in this context.

Findings

Density of $\u2208$ with $\u2208_K[]  K = _K[1/k]$ is explicitly calculated.

The density of $$ with $_K[]  K = _K$ is rac{\u03b6(n+1)}{\u03b6(n)}.

Events related to prime ideals in the spectrum are shown to be independent under certain conditions.

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 $n\geq 2$. Given $k\geq 1$, 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 $\mathfrak{p}_1\cap \mathbb{Z}\neq\mathfrak{p}_2\cap \mathbb{Z}$, 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 $|X(K,\gamma)|=t$.