2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119485In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2πi x, ξ}\}_{ξ\inŁ}$ for some set $Ł\subset\R^4$, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.6 pagesClassical Analysis and ODEsCombinatorics42B99, 20K01text