  The admittable Sobolev regularity is quantified for a function, $w$, which
has a zero in the $d$--dimensional torus and whose reciprocal $u=1/w$ is a
$(p,q)$--multiplier. Several aspects of this problem are addressed, including
zero--sets of positive Hausdorff dimension, matrix valued Fourier multipliers,
and non--symmetric versions of Sobolev regularity. Additionally, we make a
connection between Fourier multipliers and approximation properties of Gabor
systems and shift--invariant systems. We exploit this connection and the
results on Fourier multipliers to refine and extend versions of the Balian--Low
uncertainty principle in these settings.
