Math Conjectures

Two-dimensional Fuglede conjecture

OPENMajorCanonical finite caseProposed 1974 · Canonical special case

Canonical statement

Let Ω⊂R2\Omega\subset\mathbb R^2 be a bounded Lebesgue measurable set of positive measure. The following are equivalent: (i) there is a countable Λ⊂R2\Lambda\subset\mathbb R^2 such that {∣Ω∣−1/2e2πiλ⋅x:λ∈Λ}\{|\Omega|^{-1/2}e^{2\pi i\lambda\cdot x}:\lambda\in\Lambda\} is an orthonormal basis of L2(Ω)L^2(\Omega); (ii) there is a countable T⊂R2T\subset\mathbb R^2 such that ∑t∈T1Ω(x−t)=1\sum_{t\in T}\mathbf1_\Omega(x-t)=1 for almost every x∈R2x\in\mathbb R^2.
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Let \(\Omega\subset\mathbb R^2\) be a bounded Lebesgue measurable set of positive measure. The following are equivalent: (i) there is a countable \(\Lambda\subset\mathbb R^2\) such that \(\{|\Omega|^{-1/2}e^{2\pi i\lambda\cdot x}:\lambda\in\Lambda\}\) is an orthonormal basis of \(L^2(\Omega)\); (ii) there is a countable \(T\subset\mathbb R^2\) such that \(\sum_{t\in T}\mathbf1_\Omega(x-t)=1\) for almost every \(x\in\mathbb R^2\).
ANAL-FA-014:[Fuglede1974CommutingOperators][Tao2004FugledeCounterexample][JorgensenTian2025Fuglede]
Analysis and functional analysisDiscrete geometry

Notes

Fuglede conjectured in 1974 that a bounded measurable set Ω⊂Rd\Omega\subset\mathbb R^d of positive measure is spectral, meaning L2(Ω)L^2(\Omega) admits an orthonormal basis of exponentials, if and only if Ω\Omega tiles Rd\mathbb R^d by translations. The question arose from his study of commuting self-adjoint extensions of the partial differential operators −i∂xj-i\partial_{x_j} on a domain [Fuglede1974CommutingOperators]. This record concerns the two-dimensional case, now regarded as the canonical geometric case left open.

In high dimensions the conjecture is false: Tao constructed a spectral set that does not tile, via a finite-group counterexample lifted to Euclidean space [Tao2004FugledeCounterexample], and refinements of this method disproved both implications in every dimension d≥3d\ge3. The known counterexamples are essentially arithmetic, and no analogous construction is available in the plane. Connections back to the original operator-theoretic setting continue to be developed [JorgensenTian2025Fuglede].

Resolving the planar case would require either a genuinely low-dimensional proof of the tiling–spectrality equivalence or a counterexample of a kind the current arithmetic methods cannot produce; in dimensions one and two, both implications remain open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.