Strong Uniformity for Integral Points on an Algebraic Torus over Torsion Divisors

Recall Faltings' and Vojta's results on the Zariski nondensity of integral points on a semiabelian variety defined over a number field. We will consider this property in the case of $\mathbb{G}_{\mathrm{m}}^{N}$ ($N \geq 1$). If $K$ is a number field, $S$ is a finite set of places of $K$, including all the infinite ones, and $D$ is a nonzero effective divisor on $\mathbb{G}_{\mathrm{m}}^{N}$, then the set $\mathbb{G}^{N}_{\mathrm{m},D}(\mathcal{O}_{K,S})$ of $(D,S)$-integral points on $\mathbb{G}_{\mathrm{m}}^{N}$, belonging to $\mathbb{G}_{\mathrm{m}}^{N}(K)$, is not Zariski dense in $\mathbb{G}^{N}_{\mathrm{m}}$. A question is how this set varies with $D$ varying. We will prove that the union of these sets is still not Zariski dense in $\mathbb{G}^{N}_{\mathrm{m}}$ when $D$ runs over the set of all torsion hypersurfaces of bounded degree of $\mathbb{G}_{\mathrm{m}}^{N}$.