SPECTRAL PAIRS

TERENCE TAO

1. Introduction

Let D, Ω be two subsets of R^{n} of measure 1. We say that (D, Ω) are an spectral pair if

(D − D) ∩ Z_{Ω} = (Ω − Ω) ∩ Z_{D} = {0}  (1)

where Z_{Ω} denotes the zero set of χ̂_{Ω}.

The following result of Kolountzakis relates the tiling properties of D with the spectral properties of Ω.

Theorem 1.1. [2] Let (D, Ω) be a spectral pair, and Λ be a discrete subset. Let δ_{Λ} be the sum of the delta functions of Λ. Then

χ_{D} ∗ δ_{Λ} = 1  (2)

is equivalent to

|χ̂_{Ω}|^{2} ∗ δ_{Λ} = 1.  (3)

The equation (2) states that the translates of D by Λ tile the plane, while (3) is equivalent to the statement that the exponential waves given by Λ form an orthonormal basis for Ω.

Proof Suppose (2) held. By a Fourier transform we have

χ̂_{D}δ̂_{Λ} = δ;

By (1), the support of χ_{Ω} ∗ χ_{−Ω} is contained in the support of χ̂_{D}. Since both functions are continuous and equal 1 at the origin, we thus have

(χ_{Ω} ∗ χ_{−Ω})δ̂_{Λ} = δ.

Applying another Fourier transform we obtain (3).

Now suppose (3) held. By evaluating (3) on elements of Λ, and using the fact that |χ̂_{Ω}|^{2} is non-negative and equals 1 at the origin, we see that

|χ̂_{Ω}|^{2} vanishes on (Λ − Λ)\{0},

so that

Λ − Λ ⊆ Z_{Ω} ∪ {0}.

By (1), we thus have

(D − D) ∩ (Λ − Λ) = {0}.

This is equivalent to

χ_{D} ∗ δ_{Λ}(x) ≤ 1 for all x ∈ R^{n},

or equivalently

(1 − χ_{D} ∗ δ_{Λ}) ≥ 0.

On the other hand, by convolving (3) with χ_{D}, then observing from Plancherel that |χ̂_{Ω}|^{2} has unit mass, we have

(1 − χ_{D} ∗ δ_{Λ}) ∗ |χ̂_{Ω}|^{2} = 0.

Since |χ̂_{Ω}|^{2} is non-negative and is positive near the origin, we thus have

1 − χ_{D} ∗ δ_{Λ} = 0,

which is (2).

Let Q be the open unit cube centered at the origin. Since (Q, Q) is a spectral pair, this result implies the one in [1].

References

[1] A. Iosevitch, S. Pedersen, Spectral and Tiling properties of the unit cube, IMRN 16, 819-828.

[2] M. Kolountzakis, Packing, tiling, orthogonality, and completeness, preprint.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu