Fuglede's conjecture
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
- .
- S2CID 119153862.
- .
- S2CID 8267263.
- S2CID 15553212.
- Bibcode:2004math......6127K.
- .
- Bibcode:2004math.....11512K.
- )
- S2CID 55748258.
- S2CID 139105387.