Three problems on exponential bases

We consider three special and significant cases of  the following problem: let D ⊂ R^d be a (possibly unbounded) set of finite Lebesgue measure. Let E(Z^d ) = {e (2πix.n)}, with n∈ Z^d, be the standard exponential basis on the unit cube of R^d .   Find conditions on the domain D for which E(Z^d ) is a frame, or a Riesz sequence, or a Riesz basis on L^2 (D). This is a joint work with A. Mizrahi and A. Tepper