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

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

Funding

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

Funders	Funder number
HORIZON EUROPE European Research Council	713927
European Commission
Israel Science Foundation	1044/21
University of Crete	4725
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal
Hellenic Foundation for Research and Innovation	HFRI-FM17-1733
National Research, Development and Innovation Office	K129335, K132097

Keywords

Equidecomposability
Fuglede’s conjecture
Polytopes
Spectral set
Tiling

Access to Document

10.1007/s43670-023-00070-w

Cite this

@article{bad13d9e7a214203be3a22b6a71a9115,

title = "Spectral sets and weak tiling",

abstract = "A set Ω ⊂ Rd 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 Rd . 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.",

keywords = "Equidecomposability, Fuglede{\textquoteright}s conjecture, Polytopes, Spectral set, Tiling",

author = "Kolountzakis, {Mihail N.} and Nir Lev and M{\'a}t{\'e} Matolcsi",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

year = "2023",

month = dec,

doi = "10.1007/s43670-023-00070-w",

language = "אנגלית",

volume = "21",

journal = "Sampling Theory, Signal Processing, and Data Analysis",

issn = "2730-5716",

publisher = "Springer Nature Switzerland AG",

number = "2",

}

TY - JOUR

T1 - Spectral sets and weak tiling

AU - Kolountzakis, Mihail N.

AU - Lev, Nir

AU - Matolcsi, Máté

PY - 2023/12

Y1 - 2023/12

N2 - A set Ω ⊂ Rd 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 Rd . 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.

AB - A set Ω ⊂ Rd 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 Rd . 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.

KW - Equidecomposability

KW - Fuglede’s conjecture

KW - Polytopes

KW - Spectral set

KW - Tiling

UR - http://www.scopus.com/inward/record.url?scp=85174213098&partnerID=8YFLogxK

U2 - 10.1007/s43670-023-00070-w

DO - 10.1007/s43670-023-00070-w

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85174213098

SN - 2730-5716

VL - 21

JO - Sampling Theory, Signal Processing, and Data Analysis

JF - Sampling Theory, Signal Processing, and Data Analysis

IS - 2

M1 - 31

ER -

Spectral sets and weak tiling

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this