Packing, tiling, orthogonality and completeness

Abstract

Let $Ω\subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $Ω$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator function of $Ω$ and $D$ has measure 1. Suppose that $Λ$ 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 $Ω$ being the cube, by Iosevich and Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $Λ$ if and only if the set of exponentials $E_Λ= \{\exp 2πi λ\cdot x: λ\inΛ\}$ is an orthonormal basis for $L^2(Ω)$. (When $Ω$ is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of itself.) In our approach orthogonality of $E_Λ$ is viewed as a statement about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function and, additionally, we have completeness of $E_Λ$ in $L^2(Ω)$ 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.