C A ] 8 A pr 2 00 1 FOURIER BASES AND A DISTANCE PROBLEM OF ERDŐS

We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in R is at least Cdn 1 d +ǫd , ǫd > 0. Introduction and statement of results Fourier bases. Let D be a domain in R, i.e., D is a Lebesgue measurable subset of R d with finite non-zero Lebesgue measure. We say that D is a spectral set if L(D) has orthogonal basis of the form EΛ = {e}λ∈Λ, where Λ is an infinite subset of R. 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 R if ∪t∈T (D + t) is a partition of R up to 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 R 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 ∈ R : |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. We say 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 consequence of [Fug] and Theorem 1 we have: 1991 Mathematics Subject Classification. 42B. Research supported in part by NSF grants DMS97-06825 and DMS-9801410 Typeset by AMS-TEX 1 2 ALEX IOSEVICH, NETS KATZ, AND STEEN PEDERSEN Corollary. There do not exist commuting self-adjoint operators Hj acting on L (Bd) such that Hjf = − √ −1 ∂f/∂xj for f in the domain of the unbounded operator Hj and 1 ≤ j ≤ d. The derivatives ∂/∂xj act on L (Bd) in the distribution sense. In other words, there do not exist commuting self-adjoint restrictions of the partial derivative operators − √ −1 ∂/∂xj, j = 1, . , d, acting on L(Bd) in the distribution 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], Theorem 12.13. Theorem 2. Let gd(n), d ≥ 2, denote the minimum number of distances determined by n points in R. Then (*) gd(n) ≥ Cdn 3 3d−2 . Remark. The study of the problem addressed in Theorem 2 was initiated by Erdős. He proved that g2(n) ≥ Cn 1 2 . See [Erd]. Moser proved in [Mos] that g2(n) ≥ Cn 2 3 . More recently, Chung, Szeremedi, and Trotter proved that g2(n) ≥ C n 4 5 log(n) for some c > 0. See [CST]. Theorem 2 above is proved by induction using the g2(n) ≥ Cn 3 4 result proved by Clarkson et al. in [C]. As the reader shall see, Theorem 1 does not require the full strength of Theorem 2. We just need the fact gd(n) ≥ Cdn 1 d , for some ǫ > 0. 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 this was established in [JoPe2]), that Λ is a spectrum for [0, 1] d , in the sense defined above, if and only if Λ is a tiling set for [0, 1] d , in the sense that [0, 1] d + Λ = R without overlaps. It follows that [0, 1] d has lots of spectra. The standard integer lattice Λ = Z is an example, though there are many non-trivial examples as well. 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)} ≈ R, where Bd(R) denotes a ball of radius R and f(R) ≈ g(R) means that there exist 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 between the elements of {Λ ∩ Bd(R)} is ≈ R. Theorem 2 implies that if R is sufficiently large, this is not possible. Kolountzakis ([Kol]) recently proved that if D is any convex non-symmetric domain in R , then D is not a spectral set. Theorem 1 is a step in the direction of proving that if D is a convex domain such that ∂D has at least one point where the Gaussian curvature does not vanish, then D is not a spectral set. This, in its turn, would be a step towards proving the conjecture of Fuglede mentioned above.