Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and any subvariety $X \subset A$ of an abelian variety $A/K$, a $K$-rational point in $y \in A(K)$ not lying in $X$ must be at least of $v$-adic distance $e^{-o(h(y))}$ apart from $X$. Here $h : A(\bar{K}) \to \mathbb{R}$ is any fixed Weil height in the class of a symmetric projective embedding; for example, one may take a Neron-Tate height.
This corresponds to Roth's theorem, which would have $\mathbb{P}^1$ in place of $A$, and $X = \{\alpha\} \in \mathbb{P}^1(\bar{K})$ an algebraic point -- except that the exponent is then $\exp{\big(-(2+o(1))h(y)\big)}$. The common wisdom is that the exponents $0$ and $2$ correspond to the respective canonical classes of the ambient varieties $A$ and $\mathbb{P}^1$.
But one could look at this differently by taking into account the finite generation of the group $A(K)$ (Mordell-Weil): there is simply much less freedom for the rational approximants, hence the result on bad approximability must be much stronger than in $\mathbb{P}^1$. This leads one to ask if the exponent $2+o(1)$ in Roth's theorem be could be lowered to $o(1)$ if we restrict to approximations taken only from a given finitely generated subgroup $\Gamma \subset \mathbb{G}_m(\bar{K})$, and indeed one sees that this is so by Baker's theorem (on logarithmic linear forms).
Hence my question:
Is this known to be true (has it been worked out) for arbitrary subvarieties of linear tori of higher dimension? That is: given $X \subset \mathbb{G}_m^r$ a subvariety, $v$ a place of $K$, and $\Gamma \subset \mathbb{G}_m^r(K)$ a finitely generated group, is the $v$-adic distance from a point $y \in \Gamma$ to $X$ bounded below by $e^{-o(h(y))}$, unless $y \in X$?
[Added: I apologize, I realized right after asking the question that this should follow from the Faltings-Wustholz theorem (which proves bad approximability by $K$-rational points with the exponent $r+1+o(1)$), after taking the inverse images of the original data under isogenies $\rho$ of $\mathbb{G}_m^r$. Indeed, the distance from $\rho^{-1}(y)$ to $\rho^{-1}(X)$ stays about the same at some place dividing $v$, while the height of the points above $y$ gets divided by $\deg{\rho}$. Since also $\rho^{-1}(\Gamma) \subset \mathbb{G}_m^r(L)$ for some number field $L = L(\rho)$, and because we may take $\deg{\rho}$ arbitrarily large, this allows us to reduce the exponent $n+1+o(1)$ from Faltings-Wustholz to any $\epsilon > 0$ in our situation.
Still, maybe I should leaves this question rather than promptly delete it, as there might be other references or different (in particular, easier) ways to get the statement. Those I would very much appreciate. ]
