Iosevich.tex.4_8_99

Mathematical Research Letters 6, 251–255 (1999)
FOURIER BASES AND A DISTANCE PROBLEM OF ERD ˝
Alex Iosevich, Nets Katz, and Steen Pedersen We prove that no ball admits a non-harmonic orthogonal basis of ex- ponentials. We use a combinatorial result, originally studied by Erd˝ that the number of distances determined by n points in Rd is at least C Introduction and statement of results
Fourier bases. Let D be a domain in Rd, i.e., D is a Lebesgue measurable
subset of Rd with finite non-zero Lebesgue measure. We say that D is a spectral
set if L2(D) has orthogonal basis of the form EΛ = {e2πix·λ}
infinite subset of Rd. We shall refer to Λ as a spectrum for D.
We say that a family D + t, t ∈ T , of translates of a domain D tiles Rd if ∪t∈T (D + t) is a partition of Rd upto sets of Lebesgue measure zero.
Conjecture. It has been conjectured (see [Fug]) that a domain D is a spectral
set if and only if it is possible to tile Rd by a family of translates of D.
This conjecture is nowhere near resolution, even in dimension one. It has been the subject of recent research, see for example [JoPe2], [LaWa], and [Ped].
In this paper we address the following special case of the conjecture. Let Bd = {x ∈ Rd : |x| ≤ 1} denote the unit ball. We prove that Theorem 1. An affine image of D = Bd, d ≥ 2, is not a spectral set.
If A is a (possibly unbounded) self-adjoint operator acting on some Hilbert space, then we may define exp − −1A using the Spectral Theorem. Wesay that two (unbounded) self-adjoint operators A and B acting on the same Hilbert space commute if the bounded unitary operators exp − −1sA and exp − −1tB commute for all real numbers s and t. See, for example, [ReSi]for more details on the needed operator theory. As an immediate consequenceof [Fug] and Theorem 1 we have: Received March 1, 1999.
1991 Mathematics Subject Classification 42B.
Research supported in part by NSF grants DMS97-06825 and DMS-9801410.
ALEX IOSEVICH, NETS KATZ, AND STEEN PEDERSEN Corollary. There do not exist commuting self-adjoint operators Hj acting on
L2(Bd) such that Hjf = − −1 ∂f/∂xj for f in the domain of the unboundedoperator Hj and 1 ≤ j ≤ d. The derivatives ∂/∂xj act on L2(Bd) in the distri-bution sense. In other words, there do not exist commuting self-adjoint restrictions of the partial derivative operators − −1 ∂/∂xj, j = 1, . . . , d, acting on L2(Bd) in thedistribution sense.
The two-dimensional case of Theorem 1 was proved by Fuglede in [Fug]. Our proof uses the following combinatorial result. See for example [AgPa], Theorem12.13.
Theorem 2. Let gd(n), d ≥ 2, denote the minimum number of distances deter-
mined by n points in Rd. Then
gd(n) ≥ Cdn 3d−2 . Remark. The study of the problem addressed in Theorem 2 was initiated by He proved that g2(n) ≥ Cn 2 . See [Erd]. Moser proved in [Mos] that g2(n) ≥ Cn 3 . More recently, Chung, Szeremedi, and Trotter proved that for some c > 0. See [CST]. Theorem 2 above is proved by induction using the g2(n) ≥ Cn 4 result proved by Clarkson et al. in [C].
As the reader shall see, Theorem 1 does not require the full strength of The- It is interesting to contrast the case of the ball with the case of the cube [0, 1]d. It was proved in [IoPe1], (and, independently, in [LRW]; for d ≤ 3 thiswas established in [JoPe2]), that Λ is a spectrum for [0, 1]d, in the sense definedabove, if and only if Λ is a tiling set for [0, 1]d, in the sense that [0, 1]d + Λ = Rdwithout overlaps. It follows that [0, 1]d has lots of spectra. The standard integerlattice Λ = Zd is an example, though there are many non-trivial examples aswell. See [IoPe1] and [LaSh].
Our method of proof is as follows. We shall argue that if Bd were a spectral set, then any corresponding spectrum Λ would have the property #{Λ ∩ Bd(R)} ≈Rd, where Bd(R) denotes a ball of radius R and f (R) ≈ g(R) means that thereexist constants c ≤ C so that c f(R) ≤ g(R) ≤ C f(R) for R sufficiently large.
On the other hand, we will show that the number of distinct distances betweenthe elements of {Λ ∩ Bd(R)} is ≈ R. Theorem 2 implies that if R is sufficientlylarge, this is not possible.
Kolountzakis ([Kol]) recently proved that if D is any convex non-symmetric domain in Rd, then D is not a spectral set. Theorem 1 is a step in the direction ofproving that if D is a convex domain such that ∂D has at least one point where FOURIER BASES AND A DISTANCE PROBLEM OF ERD ˝ the Gaussian curvature does not vanish, then D is not a spectral set. This, inits turn, would be a steptowards proving the conjecture of Fuglede mentionedabove.
Orthogonality
ZD = {ξ ∈ R : χD(ξ) = 0}. Consider a set of exponentials EΛ. Observe that eλ(x)eλ (x) dx. It follows that the exponentials EΛ are orthogonal in L2(D) if and only if Proposition 1. If EΛ is an orthogonal subset of L2(D) then there exists a
constant C depending onlyon D such that
# (Λ ∩ Bd(R)) ≤ C Rd, for anyball Bd(R) of radius R in Rd. Proof. Since χD is continuous and χD(0) = |D| it follows that inf{|ξ| : χD(ξ) = 0} = r > 0. If ξ1, . . . , ξn are in Λ∩Bd(R) then the balls B(ξj, r/2) are disjoint and containedin Bd(R + r/2). Since r only depends on D the desired inequality follows.
To study the exact possibilities for sets Λ so that EΛ is orthogonal it is of interest to us to compute the set ZD. We will without loss of generality assumethat 0 ∈ Λ. We again compare the sets ZD for the cases where D is the cubeand the ball.
Let Qd = [0, 1]d be the cube in Rd. The zero set ZQ for χQ is the union of the hyperplanes {x ∈ Rd : xi = z}, where the union is taken over 1 ≤ i ≤ d, andover all non-zero integers z.
Let Bd = {x ∈ Rd : x ≤ 1} be the unit ball in Rd. The zero set ZB for χ is the union of the spheres {x ∈ Rd : x = r}, where the union is over all thepositive roots r of an appropriate Bessel function.
For the cube Qd it is easy to find a large set Λ ZQ ∪ {0} so that Λ − Λ ZQ ∪{0}. For example, we may take Λ = Zd. In the case of the ball B ZB ∪ {0} satisfy Λ − Λ ALEX IOSEVICH, NETS KATZ, AND STEEN PEDERSEN Proof of Theorem 1
Theorem 3. Suppose that D is a spectral set and that Λ is a spectrum for D
in the sense defined above, where D is a bounded domain. There exists an r > 0
so that anyball of radius r contains at least one point from Λ.
Proof. This is a special case of [IoPe2]. See also [Beu], [Lan], and [GrRa].
It is a consequence of Theorem 3 that if D is a spectral set then there exists a constant C > 0 such that if Λ is a spectrum for D then #{Λ ∩ Bd(R)} ≥ C Rdfor any ball Bd(R) of radius R provided that R is sufficiently large. Combiningthis with Proposition 1 we see that #{Λ ∩ Bd(R)} ≈ Rd.
Suppose Λ is a spectrum for the unit ball Bd centered at the origin in Rd. Let Bd(R) be a ball of radius R. Since #{Λ ∩ Bd(R)} ≈ Rd it follows from Theorem2 that #{|λ − λ | : λ, λ ∈ Λ ∩ Bd(R)} ≥ C R 3d−2 . Now, since χB is an analytic radial function, it follows that if f is given by f (|ξ|) = χB (ξ), then the number of zeros of f in the interval [−R, R] is bounded above by a multiple of R. In fact an explicit calculation shows that χB (ξ) = |ξ|d2 Jd (2π|ξ|), where Jν denotes the usual Bessel function of order ν. See, for f (|λ − λ |) = χB (λ − λ ) = 0. Combining the upper bound on the number of zeros of f in [−R, R] with thelower bound (**) we derived from Theorem 2 above we have C R ≥ #{|λ − λ | : λ, λ ∈ Λ ∩ Bd(R)} ≥ C R 3d−2 . this leads to a contradiction by choosing R sufficiently large.
This completes the proof of Theorem 1.
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Department of Mathematics, Georgetown University, Washington, DC 20057 E-mail address: iosevich@math.georgetown.edu Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois, Department of Mathematics, Wright State University, Dayton, OH 45435 E-mail address: steen@math.wright.edu

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