Spectra for cubes in products of finite cyclic groups

We consider ``cubes'' in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the group and is called spectral if it admits an orthogonal basis of characters for the functions supported on the set.) We show an analogue of a theorem due to Iosevich and Pedersen (1998), Lagarias, Reeds and Wang (2000), and the third author of this paper (2000), which identified the tiling complements of the unit cube in \mathbb{R}^d with the spectra of the same cube.