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arXiv · 2410.12387

Maximality and completeness of orthogonal exponentials on the cube

Mihail N. Kolountzakis·Nir Lev·Máté Matolcsi
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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.

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Mihail N. Kolountzakis, Nir Lev, Máté Matolcsi. 2024-10-16. Maximality and completeness of orthogonal exponentials on the cube. https://arxiv.org/abs/2410.12387

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