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

Abstract

We fix a number field [Formula: see text] and study statistical properties of the ring [Formula: see text] as [Formula: see text] varies over algebraic numbers of a fixed degree [Formula: see text]. Given [Formula: see text], we explicitly compute the density of [Formula: see text] for which [Formula: see text] and show that this does not depend on the number field [Formula: see text]. In particular, we show that the density of [Formula: see text] for which [Formula: see text] is [Formula: see text]. 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 [Formula: see text] to be a certain finite subset of [Formula: see text] and show that [Formula: see text] determines the ring [Formula: see text]. We show that if [Formula: see text] satisfy [Formula: see text], then the events [Formula: see text] and [Formula: see text] are independent. As [Formula: see text], we study the asymptotics of the density of [Formula: see text] for which [Formula: see text].