Packing, tiling, orthogonality and completeness

Abstract

Let $\Omega \subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $\Omega$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator function of $\Omega$ and $D$ has measure 1. Suppose that $\Lambda$ is a discrete subset of ${\bf R}^d$. The main contribution of this paper is a new way of proving the following result (proved by different methods by Lagarias, Reeds and Wang and, in the case of $\Omega$ being the cube, by Iosevich and Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $\Lambda$ if and only if the set of exponentials $E_\Lambda = \{\exp 2\pi i \lambda\cdot x: \lambda\in\Lambda\}$ is an orthonormal basis for $L^2(\Omega)$. (When $\Omega$ is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of itself.) In our approach orthogonality of $E_\Lambda$ is viewed as a statement about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function and, additionally, we have completeness of $E_\Lambda$ in $L^2(\Omega)$ if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier Analytic language and use this to prove our result.