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词条 Fuglede's conjecture
释义

  1. Spectral sets and translational tiles

  2. Partial results

  3. References

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 .
  • Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that the conjecture holds in , where is the finite group of order p.[9]

References

1. ^{{Cite journal|last=Fuglede|first=Bent|date=1974|title=Commuting self-adjoint partial differential operators and a group theoretic problem|url=|journal=J. Func. Anal|volume=16|pages=101–121|via=}}
2. ^{{Cite journal|last=Dutkay|first=Dorin Ervin|last2=Lai|first2=Chun Kit|date=2014|title=Some reduction of the spectral set conjecture to integers|url=|journal=Math. Cambridge. Proc. Soc.|volume=156|pages=123–135|via=}}
3. ^{{Cite journal|last=Iosevich|first=Alex|last2=Katz|first2=Nets|last3=Terence|first3=Tao|date=2003|title=The Fuglede spectral conjecture hold for convex planar domains|url=|journal=Math. Res. Lett.|volume=10|issue= 5-6|pages=556–569|via=}}
4. ^{{Cite journal|last=Tao|first=Terence|date=2004|title=Fuglede's conjecture is false on 5 or higher dimensions|url=|journal=Math. Res. Lett.|volume=11|issue= 2-3|pages=251–258|via=}}
5. ^{{Cite journal|last=Farkas|first=Bálint|last2=Matolcsi|first2=Máté|last3=Móra|first3=Péter|date=2006|title=On Fuglede's conjecture and the existence of universal spectra|url=|journal=J. Fourier Anal. Appl.|volume=12 |issue=5|pages=483–494|via=}}
6. ^{{Cite journal|last=Kolounzakis|first=Mihail N.|last2=Matolcsi|first2=Máté|date=2006|title=Tiles with no spectra|url=|journal=Forum Math.|volume=18 |issue=3|pages=519–528|via=}}
7. ^{{Cite journal|last=Matolcsi|first=Máté|date=2005|title=Fuglede's conjecture fails in dimension 4|url=|journal=Proc. Amer. Math. Soc.|volume=133 |issue=10|pages=3021–3026|via=}}
8. ^{{Cite journal|last=Kolounzakis|first=Mihail N.|last2=Matolcsi|first2=Máté|date=2006|title=Complex Hadamard Matrices and the spectral set conjecture|url=|journal=Collect. Math.|volume= Extra|pages=281–291|via=}}
9. ^{{Cite journal|last=Iosevich|first=Alex|last2=Mayeli|first2=Azita|last3=Pakianathan|first3=Jonathan|date=|title=The Fuglede Conjecture holds in Zp×Zp|url=https://arxiv.org/pdf/1505.00883.pdf|journal=|volume=|pages=|via=}}

1 : Conjectures
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