Northcott numbers for the house and the Weil height

For an algebraic number $alpha$ and $gammain mathbb{R}$, $h(alpha)$ 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 $Ssubset overline{mathbb{Q}}$. The Northcott number $mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $Xgeq 0$ 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 $tgeq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $tgeq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(cdot)$. (3) For all $0leq gammaleq 1$ 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 Robinons property (JR), examples that satisfy an analogue of Vidaux and Videlas 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.