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
L2(
D) has orthogonal basis of the form
EΛ =
{e2
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
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 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
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 = R
dwithout 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
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 R
d.
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 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 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 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|d2
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:
[email protected]
Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois,
Department of Mathematics, Wright State University, Dayton, OH 45435
E-mail address:
[email protected]
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Colloque franco-japonais Sept. 2oo8 Yvon Brès La mélancolie et le corps Il y a deux ans, pendant l’été 2oo6, j’avais eu l’honneur d’être invité à m’entretenir de diverses questions psychologiques et philosophiques avec des collègues de Tokyo et de Kyoto et surtout la joie d’avoir à le faire, entre autres, dans ce lieu universitaire de Todaï à Komaba où j’avai
Evolution and Human Behavior 26 (2005) 375 – 387Altruistic punishing and helping differ in sensitivity torelatedness, friendship, and future interactionsRick O’Gormana, David Sloan Wilsona,b,*, Ralph R. MillercaDepartment of Biological Sciences, Binghamton University, Binghamton, NY 13902-6000, USAbDepartment of Anthropology, Binghamton University, Binghamton, NY 13902- 6000, USAcDep