Northcott numbers for the house and the Weil height

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