PACKING, TILING, ORTHOGONALITY AND COMPLETENESS

Abstract

Let Ω⊆ℝ^d be an open set of measure 1. An open set D⊆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 ℝ^d. The main contribution of this paper is a new way of proving the following result: D tiles ℝ^d when translated at the locations Λ if and only if the set of exponentials E_Λ = {exp 2πi〈λ, x〉 [ratio ] λ ∈ Λ} is an orthonormal basis for L²(Ω). (This result has been proved by different methods by Lagarias, Reeds and Wang [9] and, in the case of Ω being the cube, by Iosevich and Pedersen [3]. When Ω is the unit cube in ℝ^d, it is a tight orthogonal packing region of itself.) In our approach, orthogonality of E_Λ is viewed as a statement about ‘packing’ ℝ^d with translates of a certain non-negative function and, additionally, we have completeness of E_Λ in L²(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.