Fuglede's conjecture in dimensions one and two

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $d\in\{1,2\}$ and let $\Omega\subset\mathbb R^d$ be a Lebesgue-measurable set with $0<|\Omega|<\infty$, without any convexity assumption. Call $\Omega$ spectral if there is a countable set $\Lambda\subset\mathbb R^d$ for which \[ \left\{|\Omega|^{-1/2}e^{2\pi i\langle\lambda,x\rangle}:\lambda\in\Lambda\right\} \] is an orthonormal basis of $L^2(\Omega)$. Call $\Omega$ a translational tile if there is a countable set $T\subset\mathbb R^d$ such that \[\sum_{t\in T}\mathbf1_\Omega(x-t)=1\quad\text{for Lebesgue-almost every }x\in\mathbb R^d.\] Here $|\Omega|$ denotes Lebesgue measure, $\langle\cdot,\cdot\rangle$ the Euclidean inner product, and $\mathbf1_\Omega$ the indicator function. These definitions concern equivalence up to null sets.

Statement recorded in the supplied source.

In dimensions one and two, is every spectral set a translational tile, and is every translational tile spectral? The question concerns the unrestricted, possibly nonconvex, version of Fuglede's conjecture. [S2]

Short English statement

Does having an orthogonal exponential basis characterize exactly the measurable sets that tile the line or plane by translations, when convexity is not assumed?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2400002 (AMR-023-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dorin Ervin Dutkay and Palle E. T. Jorgensen, The momentum operator on a union of intervals and the Fuglede conjecture (2023), arXiv:2301.11377, Abstract and Section 1, pp. 1–2; Section 6. https://arxiv.org/pdf/2301.11377.