ar X iv : m at h / 99 04 06 6 v 1 [ m at h . C A ] 1 4 A pr 1 99 9 Packing , tiling , orthogonality and completeness

Let Ω ⊆ R be an open set of measure 1. An open set D ⊆ R 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. Suppose that Λ is a discrete subset of R. The main contribution of this paper is a new way of proving the folowing result (proved by Lagarias, Reeds and Wang and, in the case of Ω being the cube, by Iosevich and Pedersen): D tiles R translated at the locations Λ if and only if the set of exponentials EΛ = {exp 2πi〈λ, x〉 : λ ∈ Λ} is an orthonormal basis for L(Ω). (When Ω is the unit cube in R then it is a tight orthogonal packing region of itself.) In our approach orthogonality of EΛ is viewed as statement about packing R d with translates of a certain nonnegative function and EΛ is an orthonormal basis for L (Ω) if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition into Fourier Analytic language and use this to prove our result. §0. Introduction Notation. Let Ω ⊂ Rd be measurable of measure 1. The Hilbert space L(Ω) is equipped with the inner product 〈f, g〉Ω = ∫ Ω f(x)g(x) dx. Define eλ(x) = exp 2πi〈λ, x〉, and for Λ ⊆ Rd define EΛ = {eλ : λ ∈ Λ}. For every continuous function h we write Z(h) = { x ∈ R : h(x) = 0 } . Whenever we fail to mention it it should be understood that the measure of Ω ⊂ Rd is equal to 1. Definition 1 (Spectral sets) Suppose that Ω is a measurable set of measure 1. We call Ω spectral if L(Ω) has an orthonormal basis EΛ = {eλ : λ ∈ Λ} of exponentials. The set Λ is then called a spectrum for Ω. We can always restrict our attention to sets Λ containing 0 and we shall do so without further mention. Partially supportedby the U.S. National Science Foundation, under grant DMS 97-05775. Most of this work was carried out while the author was visiting the Univ. of Illinois at Urbana-Champaign, in Fall 1998-99.