Spectral pairs in Cartesian coordinates

Abstract

Let $\Omega \subset R^d$ have finite positive Lebesgue measure, and let ${\mathcal{L}}^2(\Omega )$ be the corresponding Hilbert space of ${\mathcal{L}}^2$-functions on $\Omega$. We shall consider the exponential functions $e_{\lambda }$ on $\Omega$ given by $e_{\lambda }(x)=e^{i2\pi \lambda x}$. If these functions form an orthogonal basis for ${\mathcal{L}}^2(\Omega )$, when $\lambda$ ranges over some subset $\Lambda$ in $R^d$, then we say that $(\Omega ,\Lambda )$ is a spectral pair, and that $\Lambda$ is a spectrum. We conjecture that $(\Omega ,\Lambda )$ is a spectral pair if and only if the translates of some set ${\Omega }^{\prime }$ by the vectors of $\Lambda$ tile $R^d$. In the special case of $\Omega =I^d$, the $d$-dimensional unit cube, we prove this conjecture, with ${\Omega }^{\prime }=I^d$, for $d\le 3$, describing all the tilings by $I^d$, and for all $d$ when $\Lambda$ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.

AMS-LaTeX; 18 pages, 1 figure comprising 2 EPS diagrams; revision provides the graphics files for these figures (no other changes)