Separating finite Laurent support by an integer weight ($w\cdot m$)

Given a finite set $F\subset {\mathbb{Z}}^r$ of exponent vectors, there is $w\in {\mathbb{Z}}^r$ for which the integers $w\cdot m$, $m\in F$, are pairwise distinct. For $r=2$, take $w=(1,N)$ with $N$ larger than every first-coordinate difference in $F$. In higher dimensions take $w=(1,N,\dots ,N^{r-1})$ with $N$ sufficiently large. This proves that a nonzero Laurent polynomial can be detected by a substitution $X_i=T^{w_i}$ without cancellation of its distinct terms.