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## 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 R

*d *is at least

*C*
**Introduction and statement of results**
**Fourier bases. **Let

*D *be a domain in R

*d*, 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*2(

*D*) has orthogonal basis of the form

*E*Λ =

*{e*2

*πix·λ}*
inﬁnite subset of R

*d*. 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

*d *if

*∪t∈T *(

*D *+

*t*) is a partition of R

*d *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 R

*d *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

*d *:

*|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

*− −*1

*A *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

*− −*1

*sA *and
exp

*− −*1

*tB *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*
*L*2(

*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 L*2(

*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

*L*2(

*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 R

*d. Then*
*gd*(

*n*)

*≥ Cdn *3

*d−*2

*.*
*Remark. *The study of the problem addressed in Theorem 2 was initiated by
He proved that

*g*2(

*n*)

*≥ Cn *2 . See [Erd]. Moser proved in [Mos]
that

*g*2(

*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

*g*2(

*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 = R

*d*without overlaps. It follows that [0

*, *1]

*d *has lots of spectra. The standard integerlattice Λ = Z

*d *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 R

*d*, 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

*E*Λ. Observe that

*eλ*(

*x*)

*eλ *(

*x*)

*dx.*
It follows that the exponentials

*E*Λ are orthogonal in

*L*2(

*D*) if and only if

**Proposition 1. ***If E*Λ

*is an orthogonal subset of L*2(

*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 *R

*d.*
*Proof. *Since

*χD *is continuous and

*χD*(0) =

*|D| *it follows that
inf

*{|ξ| *:

*χD*(

*ξ*) = 0

*} *=

*r > *0

*.*
If

*ξ*1,

*. . . *,

*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

*E*Λ 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 R

*d*. The zero set

*ZQ *for

*χQ *is the union of
the hyperplanes

*{x ∈ *R

*d *:

*xi *=

*z}*, where the union is taken over 1

*≤ i ≤ d*, andover all non-zero integers

*z*.

Let

*Bd *=

*{x ∈ *R

*d *:

*x ≤ *1

*} *be the unit ball in R

*d*. The zero set

*ZB *for

*χ*
is the union of the spheres

*{x ∈ *R

*d *:

*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 = Z

*d*. 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 Rd*for 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 R

*d*. 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 *3

*d−*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|d*2

*Jd *(2

*π|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 *3

*d−*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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