Packing, Tiling, Ortho Preprint Gonality and Completeness

Let R d be an open set of measure 1. An open set D R d is called a tight orthogonal packing region for if D ? D does not intersect the zeros of the Fourier Transform of the indicator function of and D has measure 1. Suppose that is a discrete subset of R d. 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 being the cube, by Iosevich and Pedersen): D tiles R d when translated at the locations if and only if the set of exponentials E = fexp 2ih; xi : 2 g is an orthonormal basis for L 2 ((). (When is the unit cube in R d then it is a tight orthogonal packing region of itself.) In our approach orthogonality of E is viewed as statement about packing R d with translates of a certain nonnegative function and E is an orthonormal basis for L 2 (() 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.