Northcott numbers for the house and the Weil height

TL;DR

This paper investigates the Northcott numbers related to the house and Weil height for subrings of algebraic numbers, revealing diverse behaviors and linking these properties to decidability and conjectures in number theory.

Contribution

It characterizes possible Northcott numbers for various heights and constructs fields demonstrating different behaviors, extending known results and addressing open questions.

Findings

Every t≥1 is the Northcott number for some ring of integers w.r.t. house.

Existence of fields with Northcott numbers in [t,2t] for Weil height.

Fields with different Northcott number behaviors for weighted Weil heights, including zero and infinity.

Abstract

For an algebraic number $\alpha$ and $\gamma\in \mathbb{R}$, $h(\alpha)$ be the (logarithmic) Weil height, and $h_\gamma(\alpha)=(\mathrm{deg}\alpha)^\gamma h(\alpha)$ be the $\gamma$-weighted (logarithmic) Weil height of $\alpha$. Let $f:\overline{\mathbb{Q}}\to [0,\infty)$ be a function on the algebraic numbers $\overline{\mathbb{Q}}$, and let $S\subset \overline{\mathbb{Q}}$. The Northcott number $\mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $X\geq 0$ such that $\{\alpha \in S; f(\alpha)< X\}$ is infinite. This paper studies the set of Northcott numbers $\mathcal{N}_f(\mathcal{O})$ for subrings of $\overline{\mathbb{Q}}$ for the house, the Weil height, and the $\gamma$-weighted Weil height. We show: (1) Every $t\geq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $t\geq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(\cdot)$. (3) For all $0\leq \gamma\leq 1$ and $\gamma'<\gamma$ there exists a field $K$ with $\mathcal{N}_{h_{\gamma'}}(K)=0$ and $\mathcal{N}_{h_\gamma}(K)=\infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinon's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.