Fuglede's conjecture

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

Spectral sets and translational tiles
Spectral sets in $$\mathbb{R}^d$$

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

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

A set $$\Omega\subset\mathbb{R}^d$$ is said to tile $$\mathbb{R}^d$$ by translation (i.e. $$\Omega$$ is a translational tile) if there exist a discrete set $$\Tau$$ such that $$\bigcup_{t\in\Tau}(\Omega + t)=\mathbb{R}^d$$ and the Lebesgue measure of $$(\Omega + t) \cap (\Omega + t')$$ is zero for all $$t\neq t'$$in $$\Tau$$.

Partial results

 * Fuglede proved in 1974 that the conjecture holds if $$\Omega$$ is a fundamental domain of a lattice.
 * In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if $$\Omega$$ is a convex planar domain.
 * In 2004, Terence Tao showed that the conjecture is false on $$\mathbb{R}^{d}$$ for $$d\geq5$$. It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for $$d=3 $$ and $$4$$.   However, the conjecture remains unknown for $$d=1,2$$.
 * In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in $$\mathbb{Z}_{p}\times\mathbb{Z}_{p}$$, where $$\mathbb{Z}_{p}$$ is the cyclic group of order p.
 * In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in $$\mathbb{R}^3$$.
 * In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.