Fuglede's conjecture

Fuglege'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.^[1]

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 $\mathrm {T}$ such that $\bigcup _{t\in \mathrm {T} }(\Omega +t)=\mathbb {R} ^{d}$ and the Lebesgue measure of $(\Omega +t)\cap (\Omega +t')$ is zero for all $t\neq t'$ in $\mathrm {T}$ .^[2]

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.^[3]
In 2004, Terence Tao showed that the conjecture is false on ${\mathbb {R}}^{{d}}$ for $d\geq 5$ .^[4] 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 $4$ .^[5]^[6]^[7]^[8] However, the conjecture remains unknown for $d=1,2$ .
Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that the conjecture holds in $\mathbb {Z} _{p}\times \mathbb {Z} _{p}$ , where $\mathbb {Z} _{p}$ is the finite group of order p.^[9]

References

↑ Fuglede, Bent (1974). "Commuting self-adjoint parial differential operators and a group theoretic problem". J. Func. Anal. 16: 101–121.
↑ Dutkay, Dorin Ervin; Lai, Chun Kit (2014). "Some reduction of the spectral set conjecture to integers". Math. Cambridge. Proc. Soc. 156: 123–135.
↑ Iosevich, Alex; Katz, Nets; Terence, Tao (2003). "The Fuglede spectral conjecture hold for convex planar domains". Math. Res. Lett. 10 (5–6): 556–569.
↑ Tao, Terence (2004). "Fuglede's conjecture is false on 5 or higher dimensions". Math. Res. Lett. 11 (2–3): 251–258.
↑ Farkas, Bálint; Matolcsi, Máté; Móra, Péter (2006). "On Fuglede's conjecture and the existence of universal spectra". J. Fourier Anal. Appl. 12 (5): 483–494.
↑ Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spetra". Forum Math. 18 (3): 519–528.
↑ Matolcsi, Máté (2005). "Fuglede's conjecture fails in dimension 4". Proc. Amer. Math. Soc. 133 (10): 3021–3026.
↑ Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Complex Hadamard Matrices and the spectral set conjecture". Collect. Math. Extra: 281–291.
↑ Iosevich, Alex; MayeliJonathan Pakianathan, Azita; Pakianathan, Jonathan. "The Fuglede Conjecture holds in Zp×Zp" (PDF).

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[1] Fuglede, Bent (1974). "Commuting self-adjoint parial differential operators and a group theoretic problem". J. Func. Anal. 16: 101–121.

[2] Dutkay, Dorin Ervin; Lai, Chun Kit (2014). "Some reduction of the spectral set conjecture to integers". Math. Cambridge. Proc. Soc. 156: 123–135.

[3] Iosevich, Alex; Katz, Nets; Terence, Tao (2003). "The Fuglede spectral conjecture hold for convex planar domains". Math. Res. Lett. 10 (5–6): 556–569.

[4] Tao, Terence (2004). "Fuglede's conjecture is false on 5 or higher dimensions". Math. Res. Lett. 11 (2–3): 251–258.

[5] Farkas, Bálint; Matolcsi, Máté; Móra, Péter (2006). "On Fuglede's conjecture and the existence of universal spectra". J. Fourier Anal. Appl. 12 (5): 483–494.

[6] Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spetra". Forum Math. 18 (3): 519–528.

[7] Matolcsi, Máté (2005). "Fuglede's conjecture fails in dimension 4". Proc. Amer. Math. Soc. 133 (10): 3021–3026.

[8] Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Complex Hadamard Matrices and the spectral set conjecture". Collect. Math. Extra: 281–291.

[9] Iosevich, Alex; MayeliJonathan Pakianathan, Azita; Pakianathan, Jonathan. "The Fuglede Conjecture holds in Zp×Zp" (PDF).