In a forthcoming paper on nodal domains of Gaussian random functions, we (I and Misha Sodin) have a statement that is, roughly speaking, the following: if bounded nodal domains are possible at all, they have certain positive density. This sounds great until one asks a naive question "When are they possible at all?". Stripped of all irrelevant high tech terminology, this boils down to the following:

Let $K$ be an origin symmetric compact set in $\mathbb R^n$ having no isolated points and not contained in a hyperplane. Can one always construct a real-valued trigonometric polynomial $f(x)=\sum_{y\in K}c_y e^{i(y,x)}$ (where all but finitely many $c_y$ vanish and $c_{-y}=\bar c_y$) such that the set $f\ge 0$ has at least one bounded connected component? If not, how to describe $K$ for which it is possible?