Maximality and completeness of orthogonal exponentials on the cube

TL;DR

The paper investigates whether maximal sets of orthogonal exponentials for the unit cube must be complete. It establishes a sharp dimension-dependent dichotomy: maximal implies complete in dimensions 1 and 2, but not in dimension 3 and higher, where explicit maximal incomplete sets are constructed (both thick and thin) and extended to higher dimensions. It also shows how to lift these constructions to broader settings and proves structural properties of maximal sets, including their behavior in relation to tiling and spectrality. Additionally, the authors demonstrate that spectral sets beyond the cube can exhibit maximal incomplete orthogonal sets in 1D, illustrating limitations of a blanket Fuglede-type equivalence.

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 $1$ 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.

Theorems & Definitions (19)

• Theorem 2.1
• Lemma 2.2
• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 4.1
• Theorem 5.1
• proof
• Theorem 5.2
• proof