Let be a locally compact abelian group and let
be the Banach algebra of regular complex Borel measures on
. Given
its Fourier transform
is a continuous function defined on the Pontryagin dual of
. If the measure
is “nice” in some way then we expect some amount of regularity from the function
. For instance if
is actually an element of the subspace
of measures absolutely continuous with respect to the Haar measure of
then the Riemann-Lebesgue lemma asserts
.
The idempotent theorem of Cohen, Helson, and Rudin describes the structure of measures whose Fourier transform
takes a discrete set of values, or equivalently, since
, a finite set of values. To describe the theorem, note that we can define
for any polynomial
by taking appropriate linear combinations of convolution powers of
, and moreover we have the relation
, where on the right hand side we apply
pointwise. Thus if
takes only the values
then by setting
we obtain a decomposition of
into a linear combination of measures
whose Fourier transforms
take only values
and
. Such measures are called idempotent, because they are equivalently defined by
. By the argument just given it suffices to characterise idempotent measures: this explains the name of the theorem.
The most obvious example of an idempotent measure is the Haar measure of a compact subgroup
. Moreover we can multiply any idempotent measure
by a character
to obtain a measure
defined by
This measure will again be idempotent, as
If we add or subtract two idempotent measures then though we may not have again an idempotent measure we certainly have a measure whose Fourier transform takes integer values. On reflection, it feels more natural in the setting of harmonic analysis to require that takes values in a certain discrete subgroup than to require that it take values in
, so we relax our restriction so. The idempotent theorem states that we have already accounted for all those
such that
is integer-valued.
Theorem 1 (The idempotent theorem) For every
such that
is integer-valued there is a finite collection of compact subgroups
, characters
, and integers
such that
As a consequence we deduce a structure theorem for with
taking finitely many values, as we originally wanted: for every such
there is a finite collection of compact subgroups
, characters
, and complex numbers
such that
The theorem was first proved in the case of by Helson in 1953: in this case the theorem states simply that if
is integer-valued then it differs from some periodic function in finitely many places. In 1959 Rudin gave the theorem its present form and proved it for
. Finally in 1960 Cohen proved the general case, in the same paper in which he made the first substantial progress on the Littlewood problem. The proof was subsequently simplified a good deal, particularly by Amemiya and Ito in 1964. We reproduce their proof here.
First note that if is integer-valued then
is supported on a compact subgroup. Indeed by inner regularity there is a compact set
such that
, the set
of all
such that
on
is then open, and if
then
But if then
so for all
. Thus
is an open subgroup of
, so by Pontryagin duality its preannihilator
is a compact subgroup of
. Clearly
is supported on
. Thus from now on we assume
is compact.
Fix a measure and let
.
Proof: Fix and suppose we could find
such that
and
. Let
be close enough to
that
. Write
and
. Then if
is the complex number
, then
and
so we must have
Thus also
But if this holds for both and
, say with
, then we have
so , so
This proves the lemma.
Lemma 3 If
is a weak* limit point of
then
is singular with respect to the Haar measure
of
.
Proof: By the Radon-Nikodym theorem we have a decomposition for some
and some
singular with respect to
. By the Riemann-Lebesgue lemma then
is a limit point of
. Thus for any open set
and
such that
and
outside of
we have
so . This inequality extends to Borel sets in the usual way, so
is singular.
The theorem follows relatively painlessly from the two lemmas. Fix with
integer-valued and let
. Then
is weak* compact, so because
is lower semicontinuous in the weak* topology there is some
of minimal norm. Since
is an integer different from
we must have
. Thus by Lemma~2 the set
is finite. But this implies that
for some ,
, and
the support group of
. In particular
is absolutely continuous with respect to
, so because
is in the weak* closure of
we deduce from Lemma 2 that
for some
. Thus
is a nonzero measure of the form (1) and we have an obvious mutually singular decomposition
Since the theorem follows by induction.