Orthonormal bases of exponentials for the n-cube

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Fuglede’s conjecture fails in dimension 4 (2005, unknown): “For further recent results related to Fuglede’s conjecture, see [3], [12], [2] and [6].”
The spectral set conjecture and multiplicative properties of roots of polynomials (2000, unknown): “C”
Fourier bases and a distance problem of Erd\H os (2001, unknown): “m 2. We just need the fact gd(n) ≥ Cdn 1 d+ǫ, for some ǫ > 0. It is interesting to contrast the case of the ball with the case of the cube [0,1]d. It was proved in [IoPe1], (and, independently, in [LRW]; for d ≤ 3 this was established in [JoPe2]), that Λ is a spectrum for [0,1]d, in the sense deﬁ”
The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials (2002, unknown): “, [3], [4], [6], [8], [11], [9], [12], [13], [16], [17], [18], [19].”
Fuglede’s conjecture for a union of two intervals (2000, background): “, n [2], [7], [16]; other relations between the tiling and spectral properties of subsets of Rn have been conjectured and in some cases proved, see [6], [8], [9], [11], [12], [14].”