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
But if this holds for both and , say with , then we have
so , so
This proves the lemma.
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.