Spectral and Tiling Properties of the Unit Cube

Let Q := [0,1) denote the unit cube in d-dimensional Euclidean space R. Let T be a discrete subset of R. We say T is a tiling set for Q if each x ∈ R can be written uniquely as x = q+ t, with q ∈ Q and t ∈ T. We say T is a spectrum for Q, if the exponentials et(x) := e, t ∈ T form an orthonormal basis for L2(Q). Here, the juxtaposition tx of vectors t, x inR denotes the usual inner product tx = t1x1 + · · · + tdxd in R, and L2(Q) is equipped with the usual inner product, namely,