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.

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 .

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.
  • In 2004, Terence Tao showed that the conjecture is false on for . It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for and . 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.
  • In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in .
  • In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.