  For the $n$-dimensional multiparameter quantum torus algebra
$\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicatively
antisymmetric matrix $\mathfrak q = (q_{ij})$ we show that in the case when the
torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$ is
large enough there is a characteristic set of values (possibly with gaps) from
$0$ to $n$ that can occur as the Gelfand--Kirillov dimension of simple modules.
The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$ and
$\Lambda_{\mathfrak q}$ is simple studied in A.~Gupta, {\it
$\uppercase{\mbox{GK}}$-dimensions of simple modules over $K[X^{\pm 1},
\sigma]$}, Comm. Algebra, {\bf 41(7)} (2013), 2593--2597 is considered without
assuming simplicity and it is shown that a dichotomy still holds for the GK
dimension of simple modules.
