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

We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (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 , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]