Fuglede's conjecture

About 52 years old · traced to 1974

For $\Omega \subset \mathbb{R}^{d}$ with positive finite Lebesgue measure, say that $\Omega$ is spectral if there exists a set $\Lambda \subset \mathbb{R}^{d}$ such that

\left\{ e^{2\pi i \langle \lambda, \cdot \rangle} \right\}_{\lambda \in \Lambda}

is an orthogonal basis of $L^{2}(\Omega)$ , where $\langle \cdot, \cdot \rangle$ is the standard inner product on $\mathbb{R}^{d}$ ; such a $\Lambda$ is called a spectrum of $\Omega$ .

Say that $\Omega$ tiles $\mathbb{R}^{d}$ by translation if there exists a discrete set $T \subset \mathbb{R}^{d}$ such that

\bigcup_{t \in T} (\Omega + t) = \mathbb{R}^{d}

and $(\Omega + t) \cap (\Omega + t')$ has Lebesgue measure zero for all $t \neq t'$ in $T$ .

For every $d \geq 1$ and every $\Omega \subset \mathbb{R}^{d}$ of positive finite Lebesgue measure,

\Omega \text{ is spectral} \iff \Omega \text{ tiles } \mathbb{R}^{d} \text{ by translation}.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Fuglede's conjecture
Fuglede's conjecture is a problem in mathematics proposed by Bent Fuglede in 1974, and resolved in the negative for most dimensions by Terence Tao in 2004. 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.
source: Wikipedia

References

Primary source

Wikipedia

Additional references

Wikipedia, Fuglede's conjecture, the article this problem comes from.

analysis unsolved-problem

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A 2026 preprint claims explicit planar counterexamples, so the conjecture would be false even in two dimensions, but this claim has not been independently verified.

Bent Fuglede proposed the conjecture in 1974: spectrality and translational tiling should be equivalent for finite-measure sets in every Euclidean dimension. The unrestricted statement is now known to fail in dimensions $d \ge 3$ , while the planar case had remained open.

Known results

Fuglede proved the equivalence when the spectrum or translation set is a lattice (1974).
Iosevich, Katz, and Tao proved it for convex planar domains (2003).
Tao produced counterexamples in dimensions $d \ge 5$ (2004); later work reached $d=3,4$ .
The equivalence holds for convex bodies in all dimensions (2019).

2026 preprint: planar counterexamples

A 2026 preprint claims two explicit finite unions of unit squares in $\mathbb{R}^{2}$ : one tiles but is not spectral, and another is spectral but does not tile. If correct, this settles the stated conjecture negatively in every dimension at least $2$ ; the claim is unverified.

Current status (as of September 2026): The conjecture is disproved in dimensions $d \ge 3$ ; a 2026 preprint claims disproof in $d=2$ , while the one-dimensional case remains open.

Sources

No solutions have been posted yet.