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 } n∈Z d be the standard exponential basis on the unit cube of R d . Find conditions on D for which E(Z d ) is a frame, a Riesz sequence, or a Riesz basis for L 2 (D).
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