Maximality and completeness of orthogonal exponentials on the cube
Abstract
It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials.
Cite
@article{arxiv.2410.12387,
title = {Maximality and completeness of orthogonal exponentials on the cube},
author = {Mihail N. Kolountzakis and Nir Lev and Máté Matolcsi},
journal= {arXiv preprint arXiv:2410.12387},
year = {2025}
}
Comments
13 pages, no figures; added reference [LP11] where some of the problems were already solved