Fuglede's conjecture fails in dimension 4

In this note we modify a recent example of Tao and give an example of a set $\Omega\subset \mathbb{R} ^4$ such that $L^2(\Omega )$ admits an orthonormal basis of exponentials $\{\frac{1}{\vert\Omega \vert^{1/2}}e^{2\pi i \langle x, \xi \rangle }\}_{\xi\in\Lambda}$ for some set $\Lambda\subset\mathbb{R} ^4$, but which does not tile $\mathbb{R} ^4$ by translations. This 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.