  Recently, Nitzan and Olsen showed that Balian-Low theorems (BLTs) hold for
discrete Gabor systems defined on $\mathbb{Z}_d$. Here we extend these results
to a multivariable setting. Additionally, we show a variety of applications of
the Quantitative BLT, proving in particular nonsymmetric BLTs in both the
discrete and continuous setting for functions with more than one argument.
Finally, in direct analogy of the continuous setting, we show the Quantitative
Finite BLT implies the Finite BLT.
