The cube spectral-tiling conjecture

Let $I^{d}$ denote the $d$-dimensional unit cube, and let $L\subset\mathbb{R}^{d}$. The cube spectral-tiling conjecture. $(I^{d},L)$ is a spectral pair if and only if $(I^{d},L)$ is a tiling pair. This is the specialization of the spectral-set equivalence to the unit cube and is part of the paper's study of spectral pairs in Cartesian coordinates.