Northcott numbers for the house and the Weil height

Abstract

For an algebraic number $α$ and $γ\in \mathbb{R}$, $h(α)$ be the (logarithmic) Weil height, and $h_γ(α)=(\mathrm{deg}α)^γh(α)$ be the $γ$-weighted (logarithmic) Weil height of $α$. 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 $\{α\in S; f(α)< X\}$ 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 $γ$-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 $0\leq γ\leq 1$ and $γ'<γ$ there exists a field $K$ with $\mathcal{N}_{h_{γ'}}(K)=0$ and $\mathcal{N}_{h_γ}(K)=\infty$. 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.