Spectra for cubes in products of finite cyclic groups

TL;DR

This paper investigates the tiling and spectral properties of cubes in products of finite cyclic groups, establishing an analog of a Euclidean space theorem relating tiling complements and spectra.

Contribution

It extends known Euclidean space results to finite cyclic groups, linking tiling complements with spectra for cubes in these groups.

Findings

Established an analog of the Euclidean tiling-spectral theorem for finite cyclic groups.

Identified conditions under which cubes in these groups are spectral or tiles.

Connected tiling complements with spectra in the context of finite cyclic groups.

Abstract

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 orhogonal basis of characters for the functions supported on the set.) We show an analog of a theorem due to Iosevich and Pedersen, Lagarias, Reeds and Wang, and the third author of this paper, which identified the tiling complements of the unit cube in Euclidean space with the spectra of the same cube.