Fuglede's conjecture for a union of two intervals (Q2723494)

Let \(\Omega\) be a subset of \(\mathbb R^n\). \(\Omega\) is said to \textit{tile} if \(\mathbb R^n\) can be partitioned (modulo sets of measure zero) by translates of \(\Omega\). \(\Omega\) is said to be \textit{spectral} if \(L^2(\Omega)\) contains a complete orthonormal basis consisting of (normalized) plane waves \(\exp(2 pi \xi \cdot x)/|\Omega|^{1/2}\). Fuglede's conjecture asserts that a set tiles if and only if it is spectral. This conjecture remains widely open, mainly because of formidable algebraic obstacles, although there has been much recent progress in the case that \(\Omega\) is convex. In this paper the author proves the conjecture is true for the simplest non-convex case, namely sets which are the union of two intervals. Indeed, for this case the explicit tilings and the explicit spectra are given. As in previous work, one of the key tools is an explicit computation of the zero set of the Fourier transform of \(\chi_\Omega\).