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Fuglede's conjecture
Mathematical problem

Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of R d {\displaystyle \mathbb {R} ^{d}} (i.e. subset of R d {\displaystyle \mathbb {R} ^{d}} with positive finite Lebesgue measure) is a spectral set if and only if it tiles R d {\displaystyle \mathbb {R} ^{d}} by translation.

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Spectral sets and translational tiles

Spectral sets in R d {\displaystyle \mathbb {R} ^{d}}

A set Ω {\displaystyle \Omega } ⊂ {\displaystyle \subset } R d {\displaystyle \mathbb {R} ^{d}} with positive finite Lebesgue measure is said to be a spectral set if there exists a Λ {\displaystyle \Lambda } ⊂ {\displaystyle \subset } R d {\displaystyle \mathbb {R} ^{d}} such that { e 2 π i ⟨ λ , ⋅ ⟩ } λ ∈ Λ {\displaystyle \left\{e^{2\pi i\left\langle \lambda ,\cdot \right\rangle }\right\}_{\lambda \in \Lambda }} is an orthogonal basis of L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} . The set Λ {\displaystyle \Lambda } is then said to be a spectrum of Ω {\displaystyle \Omega } and ( Ω , Λ ) {\displaystyle (\Omega ,\Lambda )} is called a spectral pair.

Translational tiles of R d {\displaystyle \mathbb {R} ^{d}}

A set Ω ⊂ R d {\displaystyle \Omega \subset \mathbb {R} ^{d}} is said to tile R d {\displaystyle \mathbb {R} ^{d}} by translation (i.e. Ω {\displaystyle \Omega } is a translational tile) if there exist a discrete set T {\displaystyle \mathrm {T} } such that ⋃ t ∈ T ( Ω + t ) = R d {\displaystyle \bigcup _{t\in \mathrm {T} }(\Omega +t)=\mathbb {R} ^{d}} and the Lebesgue measure of ( Ω + t ) ∩ ( Ω + t ′ ) {\displaystyle (\Omega +t)\cap (\Omega +t')} is zero for all t ≠ t ′ {\displaystyle t\neq t'} in T {\displaystyle \mathrm {T} } .2

Partial results

  • Fuglede proved in 1974 that the conjecture holds if Ω {\displaystyle \Omega } is a fundamental domain of a lattice.
  • In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if Ω {\displaystyle \Omega } is a convex planar domain.3
  • In 2004, Terence Tao showed that the conjecture is false on R d {\displaystyle \mathbb {R} ^{d}} for d ≥ 5 {\displaystyle d\geq 5} .4 It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for d = 3 {\displaystyle d=3} and 4 {\displaystyle 4} .5678 However, the conjecture remains unknown for d = 1 , 2 {\displaystyle d=1,2} .
  • In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in Z p × Z p {\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}} , where Z p {\displaystyle \mathbb {Z} _{p}} is the cyclic group of order p.9
  • In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in R 3 {\displaystyle \mathbb {R} ^{3}} .10
  • In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.11

References

  1. Fuglede, Bent (1974). "Commuting self-adjoint partial differential operators and a group theoretic problem". Journal of Functional Analalysis. 16: 101–121. doi:10.1016/0022-1236(74)90072-X. /wiki/Doi_(identifier) ↩

  2. Dutkay, Dorin Ervin; Lai, Chun–Kit (2014). "Some reductions of the spectral set conjecture to integers". Mathematical Proceedings of the Cambridge Philosophical Society. 156 (1): 123–135. arXiv:1301.0814. Bibcode:2014MPCPS.156..123D. doi:10.1017/S0305004113000558. S2CID 119153862. /wiki/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society ↩

  3. Iosevich, Alex; Katz, Nets; Terence, Tao (2003). "The Fuglede spectral conjecture hold for convex planar domains". Mathematical Research Letters. 10 (5–6): 556–569. doi:10.4310/MRL.2003.v10.n5.a1. https://doi.org/10.4310%2FMRL.2003.v10.n5.a1 ↩

  4. Tao, Terence (2004). "Fuglede's conjecture is false on 5 or higher dimensions". Mathematical Research Letters. 11 (2–3): 251–258. arXiv:math/0306134. doi:10.4310/MRL.2004.v11.n2.a8. S2CID 8267263. /wiki/ArXiv_(identifier) ↩

  5. Farkas, Bálint; Matolcsi, Máté; Móra, Péter (2006). "On Fuglede's conjecture and the existence of universal spectra". Journal of Fourier Analysis and Applications. 12 (5): 483–494. arXiv:math/0612016. Bibcode:2006math.....12016F. doi:10.1007/s00041-005-5069-7. S2CID 15553212. /wiki/ArXiv_(identifier) ↩

  6. Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spectra". Forum Mathematicum. 18 (3): 519–528. arXiv:math/0406127. Bibcode:2004math......6127K. /wiki/ArXiv_(identifier) ↩

  7. Matolcsi, Máté (2005). "Fuglede's conjecture fails in dimension 4". Proceedings of the American Mathematical Society. 133 (10): 3021–3026. doi:10.1090/S0002-9939-05-07874-3. https://doi.org/10.1090%2FS0002-9939-05-07874-3 ↩

  8. Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Complex Hadamard Matrices and the spectral set conjecture". Collectanea Mathematica. Extra: 281–291. arXiv:math/0411512. Bibcode:2004math.....11512K. /wiki/ArXiv_(identifier) ↩

  9. Iosevich, Alex; Mayeli, Azita; Pakianathan, Jonathan (2017). "The Fuglede Conjecture holds in Zp×Zp". Analysis & PDE. 10 (4): 757–764. arXiv:1505.00883. doi:10.2140/apde.2017.10.757. /wiki/ArXiv_(identifier) ↩

  10. Greenfeld, Rachel; Lev, Nir (2017). "Fuglede's spectral set conjecture for convex polytopes". Analysis & PDE. 10 (6): 1497–1538. arXiv:1602.08854. doi:10.2140/apde.2017.10.1497. S2CID 55748258. /wiki/ArXiv_(identifier) ↩

  11. Lev, Nir; Matolcsi, Máté (2022). "The Fuglede conjecture for convex domains is true in all dimensions". Acta Mathematica. 228 (2): 385–420. arXiv:1904.12262. doi:10.4310/ACTA.2022.v228.n2.a3. S2CID 139105387. /wiki/ArXiv_(identifier) ↩