Simultaneous $\mathfrak{p}$-orderings and equidistribution

TL;DR

This paper explores the concept of simultaneous -orderings in Dedekind domains, their existence, and connections to integer-valued polynomials, highlighting recent progress and open questions in the area.

Contribution

It provides an overview of the progress on characterizing subsets of Dedekind domains that admit simultaneous -orderings and discusses related open problems.

Findings

Summarizes recent advances in simultaneous -orderings

Links the concept to integer-valued polynomial theory

Lists open problems in the field

Abstract

Let $D$ be a Dedekind domain. Roughly speaking, a simultaneous $\mathfrak{p}$-ordering is a sequence of elements from $D$ which is equidistributed modulo every power of every prime ideal in $D$ as well as possible. Bhargava asked which subsets of the Dedekind domains admit simultaneous $\mathfrak{p}$-orderings. We give an overview on the progress in this problem. We also explain how it relates to the theory of integer valued polynomials and list some open problems.