This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.

A nice remark by Christian Remling is that choosing separable functions yields that both functions can vanish on a half-space. https://mathoverflow.net/questions/482613/can-a-function-and-its-fourier-transform-both-vanish-on-a-convex-cone? The theorem by Shapiro mentioned there says that $A$ and $B$ cannot be resp. a "major convex cone" (strictly larger than a half space) and a non-empty open set.

Hence a remaining question is what can happen if $A$ is a "minor cone", say the quadrant of points with non-negative coordinates. As we saw, $B$ can be a half-space, but not a major cone. Is it possible that $B$ has a "gap in each coordinate"? 

Here is a one-sentence question: is it possible to find a non-null Schwartz function $f$ on $\mathbb R^2$ which vanishes on a quarter-space and which Fourier transform $\hat f(x,y)$ vanishes for $x\in O$ or $y\in O'$ where $O,O'$ are non-empty open sets of $\mathbb R$.