Let Ω⊆ℝd be an open set of measure 1. An open set
D⊆ℝd 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 ℝd. The main contribution of this paper is
a new way of proving the following result: D tiles ℝd when
translated at the locations Λ if and only if the set of exponentials
EΛ = {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 ℝd, it is a tight orthogonal packing region of
itself.) In our approach, orthogonality of EΛ is viewed as a statement
about ‘packing’ ℝd with translates
of a certain non-negative function and, additionally, we have completeness of
EΛ 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.