Spectral sets and weak tiling

Mihail N. Kolountzakis; Nir Lev; Máté Matolcsi

doi:10.1007/s43670-023-00070-w

Spectral sets and weak tiling

Mihail N. Kolountzakis, Nir Lev, Máté Matolcsi

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A set Ω ⊂ R^d is said to be spectral if the space L²(Ω) admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that Ω is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in R^d . The proof was based on a new geometric necessary condition for spectrality, called “weak tiling”. In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.

Original language	English
Article number	31
Journal	Sampling Theory, Signal Processing, and Data Analysis
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s43670-023-00070-w
State	Published - Dec 2023

Bibliographical note

Funding Information:
M.K. was supported by the Hellenic Foundation for Research and Innovation, Project HFRI-FM17-1733 and by University of Crete Grant 4725. N.L. was supported by ISF Grant No. 1044/21 and ERC Starting Grant No. 713927. M.M. was supported by NKFIH Grants K129335 and K132097

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

Equidecomposability
Fuglede’s conjecture
Polytopes
Spectral set
Tiling

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10.1007/s43670-023-00070-w

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Spectral sets and weak tiling

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