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Fuglede's conjecture

From Wikipedia, the free encyclopedia

Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of (i.e. subset of with positive finite Lebesgue measure) is a spectral set if and only if it tiles by translation.[1]

Spectral sets and translational tiles

Spectral sets in

A set with positive finite Lebesgue measure is said to be a spectral set if there exists a such that is an orthogonal basis of . The set is then said to be a spectrum of and is called a spectral pair.

Translational tiles of

A set is said to tile by translation (i.e. is a translational tile) if there exist a discrete set such that and the Lebesgue measure of is zero for all in .[2]

Partial results

  • Fuglede proved in 1974 that the conjecture holds if is a fundamental domain of a lattice.
  • In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if is a convex planar domain.[3]
  • In 2004, Terence Tao showed that the conjecture is false on for .[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for and .[5][6][7][8] However, the conjecture remains unknown for .
  • In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in , where is the cyclic group of order p.[9]
  • In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in .[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". J. Funct. Anal. 16: 101–121. doi:10.1016/0022-1236(74)90072-X.
  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.
  3. ^ Iosevich, Alex; Katz, Nets; Terence, Tao (2003). "The Fuglede spectral conjecture hold for convex planar domains". Math. Res. Lett. 10 (5–6): 556–569. doi:10.4310/MRL.2003.v10.n5.a1.
  4. ^ Tao, Terence (2004). "Fuglede's conjecture is false on 5 or higher dimensions". Math. Res. Lett. 11 (2–3): 251–258. arXiv:math/0306134. doi:10.4310/MRL.2004.v11.n2.a8. S2CID 8267263.
  5. ^ Farkas, Bálint; Matolcsi, Máté; Móra, Péter (2006). "On Fuglede's conjecture and the existence of universal spectra". J. Fourier Anal. Appl. 12 (5): 483–494. arXiv:math/0612016. Bibcode:2006math.....12016F. doi:10.1007/s00041-005-5069-7. S2CID 15553212.
  6. ^ Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spectra". Forum Math. 18 (3): 519–528. arXiv:math/0406127. Bibcode:2004math......6127K.
  7. ^ Matolcsi, Máté (2005). "Fuglede's conjecture fails in dimension 4". Proc. Amer. Math. Soc. 133 (10): 3021–3026. doi:10.1090/S0002-9939-05-07874-3.
  8. ^ Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Complex Hadamard Matrices and the spectral set conjecture". Collect. Math. Extra: 281–291. arXiv:math/0411512. Bibcode:2004math.....11512K.
  9. ^ Iosevich, Alex; Mayeli, Azita; Pakianathan, Jonathan (2015). "The Fuglede Conjecture holds in Zp×Zp". arXiv:1505.00883. doi:10.2140/apde.2017.10.757. {{cite journal}}: Cite journal requires |journal= (help)
  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.
  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.
This page was last edited on 20 January 2024, at 04:21
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