Fuglede conjecture in low dimensions 1 and 2

Determine whether the Fuglede conjecture holds in Euclidean space R^n for n = 1 and n = 2; specifically, establish whether a measurable set of unit measure tiles R^n by translations if and only if it admits an orthonormal basis of exponential functions (i.e., is spectral).

Background

The Fuglede conjecture asserts an equivalence between translational tiling and spectrality for measurable sets of unit measure in Euclidean spaces. It has been disproven in higher dimensions (n ≥ 3), with explicit counterexamples constructed following Tao’s work and subsequent developments.

Despite extensive progress and many partial results, the status of the conjecture in the low-dimensional Euclidean cases n = 1 and n = 2 remains unresolved. The paper references this longstanding gap while presenting positive results in p-adic settings and related groups.

The authors’ new results concern the p-adic field Q_p and certain product groups, where they show tiling-by-function properties and spectrality, thereby highlighting the contrast with the unresolved Euclidean low-dimensional cases.