# The idempotent theorem

Let ${G}$ be a locally compact abelian group and let ${M(G)}$ be the Banach algebra of regular complex Borel measures on ${G}$. Given ${\mu\in M(G)}$ its Fourier transform

$\displaystyle \hat{\mu}(\gamma) = \int \overline{\gamma}\,d\mu,$

is a continuous function defined on the Pontryagin dual ${\hat{G}}$ of ${G}$. If the measure ${\mu}$ is “nice” in some way then we expect some amount of regularity from the function ${\hat{\mu}}$. For instance if ${\mu}$ is actually an element of the subspace ${L^1(G)\subset M(G)}$ of measures absolutely continuous with respect to the Haar measure of ${G}$ then the Riemann-Lebesgue lemma asserts ${\hat{\mu}\in C_0(\hat{G})}$.

The idempotent theorem of Cohen, Helson, and Rudin describes the structure of measures ${\mu}$ whose Fourier transform ${\hat{\mu}}$ takes a discrete set of values, or equivalently, since ${\|\hat{\mu}\|_\infty\leq\|\mu\|}$, a finite set of values. To describe the theorem, note that we can define ${P(\mu)}$ for any polynomial ${P}$ by taking appropriate linear combinations of convolution powers of ${\mu}$, and moreover we have the relation ${\widehat{P(\mu)} = P(\hat{\mu})}$, where on the right hand side we apply ${P}$ pointwise. Thus if ${\hat{\mu}}$ takes only the values ${a_1,\dots,a_n}$ then by setting

$\displaystyle P_i(x) = \prod_{j\neq i} (x-a_j)/(a_i-a_j)$

we obtain a decomposition ${\mu = a_1\mu_1 + \cdots + a_n\mu_n}$ of ${\mu}$ into a linear combination of measures ${\mu_i=P_i(\mu)}$ whose Fourier transforms ${\hat{\mu_i} = P_i(\hat{\mu})}$ take only values ${0}$ and ${1}$. Such measures are called idempotent, because they are equivalently defined by ${\mu\ast\mu=\mu}$. 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 ${m_H}$ of a compact subgroup ${H\leq G}$. Moreover we can multiply any idempotent measure ${\mu}$ by a character ${\gamma\in\hat{G}}$ to obtain a measure ${\gamma\mu}$ defined by

$\displaystyle \int f \,d(\gamma\mu) = \int f\gamma\,d\mu.$

This measure ${\gamma\mu}$ will again be idempotent, as

$\displaystyle \begin{array}{rcl} \int f\,d(\gamma \mu\ast\gamma \mu) &=& \int\int f(x+y)\gamma(x)\gamma(y)\,d\mu(x)d\mu(y) \\ &=& \int\int f(x+y) \gamma(x+y)\,d\mu(x)d\mu(y) \\ &=& \int f\gamma\,d\mu. \end{array}$

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 ${\hat{\mu}}$ takes values in a certain discrete subgroup than to require that it take values in ${\{0,1\}}$, so we relax our restriction so. The idempotent theorem states that we have already accounted for all those ${\mu}$ such that ${\hat{\mu}}$ is integer-valued.

Theorem 1 (The idempotent theorem) For every ${\mu\in M(G)}$ such that ${\hat{\mu}}$ is integer-valued there is a finite collection of compact subgroups ${G_1,\dots,G_k\leq G}$, characters ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and integers ${n_1,\dots,n_k\in\mathbf{Z}}$ such that

$\displaystyle \mu = n_1\gamma_1 m_{G_1} + \cdots + n_k\gamma_k m_{G_k}.$

As a consequence we deduce a structure theorem for ${\mu}$ with ${\hat{\mu}}$ taking finitely many values, as we originally wanted: for every such ${\mu}$ there is a finite collection of compact subgroups ${G_1,\dots,G_k\leq G}$, characters ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and complex numbers ${a_1,\dots,a_k\in\mathbf{C}}$ such that

$\displaystyle \mu = a_1\gamma_1 m_{G_1} + \cdots + a_k \gamma_k m_{G_k}.$

The theorem was first proved in the case of ${G=\mathbf{R}/\mathbf{Z}}$ by Helson in 1953: in this case the theorem states simply that if ${\hat{\mu}}$ 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 ${(\mathbf{R}/\mathbf{Z})^d}$. 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 ${\hat{\mu}}$ is integer-valued then ${\mu}$ is supported on a compact subgroup. Indeed by inner regularity there is a compact set ${K}$ such that ${|\mu|(K^c)<0.1}$, the set ${U}$ of all ${\gamma\in\hat{G}}$ such that ${|1-\gamma|<0.1/\|\mu\|}$ on ${K}$ is then open, and if ${\gamma\in U}$ then

$\displaystyle \|\gamma\mu-\mu\| = \int_G |\gamma-1|\,d|\mu| \leq \int_K + \int_{K^c} < 0.1 + 0.1 < 1.$

But if ${\gamma\mu\neq\mu}$ then

$\displaystyle \|\gamma\mu-\mu\|\geq \|\widehat{\gamma\mu}-\hat{\mu}\|_\infty \geq 1,$

so ${\gamma\mu=\mu}$ for all ${\gamma\in U}$. Thus ${\Gamma=\{\gamma\in\hat{G}: \gamma\mu=\mu\}}$ is an open subgroup of ${\hat{G}}$, so by Pontryagin duality its preannihilator ${\Gamma^\perp = \{g\in G: \gamma(g)=1 \text{ for all }\gamma\in\Gamma\}}$ is a compact subgroup of ${G}$. Clearly ${\mu}$ is supported on ${\Gamma^\perp}$. Thus from now on we assume ${G}$ is compact.

Fix a measure ${\mu\in M(G)}$ and let ${A=\{\gamma\mu: \gamma\in\hat{G}\}}$.

Lemma 2 If ${\nu}$ is a weak* limit point of ${A}$ then ${\|\nu\|<\|\mu\|}$.

Proof: Fix ${\epsilon>0}$ and suppose we could find ${f\in C(G)}$ such that ${\|f\|_\infty\leq 1}$ and ${\int f\,d\nu > (1-\epsilon)\|\mu\|}$. Let ${\gamma\mu}$ be close enough to ${\nu}$ that ${\Re\int f\gamma\,d\mu > (1-\epsilon)\|\mu\|}$. Write ${\mu = \theta|\mu|}$ and ${f\gamma\theta = g + ih}$. Then if ${Z}$ is the complex number ${Z = \int (g+i|h|)\,d|\mu|}$, then ${|Z|\leq\|\mu\|}$ and

$\displaystyle \Re Z = \int g \,d|\mu| = \Re\int f\gamma\,d\mu > (1-\epsilon)\|\mu\|,$

so we must have

$\displaystyle \Im Z = \int |h|\,d|\mu| \leq (1-(1-\epsilon)^2)^{1/2}\|\mu\| \leq 2\epsilon^{1/2}\|\mu\|.$

Thus also

$\displaystyle \int |1 - f\gamma\theta| \,d|\mu| \leq \int |1 - g|\,d|\mu| + \int |h|\,d|\mu| \leq 3\epsilon^{1/2}\|\mu\|.$

But if this holds for both ${\gamma_1\mu}$ and ${\gamma_2\mu}$, say with ${\gamma_1\mu\neq\gamma_2\mu}$, then we have

$\displaystyle 1\leq \|\gamma_1\mu-\gamma_2\mu\| \leq \int |\gamma_1 - f\gamma_1\gamma_2\theta|\,d|\mu| + \int |\gamma_2 - f\gamma_1\gamma_2\theta|\,d|\mu| \leq 6\epsilon^{1/2}\|\mu\|,$

so ${\epsilon \geq 1/(36\|\mu\|^2)}$, so

$\displaystyle \|\nu\| \leq \|\mu\| - \frac{1}{36\|\mu\|}.$

This proves the lemma. $\Box$

Lemma 3 If ${\nu}$ is a weak* limit point of ${A}$ then ${\nu}$ is singular with respect to the Haar measure ${m_G}$ of ${G}$.

Proof: By the Radon-Nikodym theorem we have a decomposition ${\mu = f m_G + \mu_s}$ for some ${f\in L^1(G)}$ and some ${\mu_s}$ singular with respect to ${m_G}$. By the Riemann-Lebesgue lemma then ${\nu}$ is a limit point of ${\{\gamma\mu_s:\gamma\in\hat{G}\}}$. Thus for any open set ${U}$ and ${f\in C(G)}$ such that ${\|f\|_\infty\leq 1}$ and ${f=0}$ outside of ${U}$ we have

$\displaystyle \left|\int f\,d\nu\right| \leq \sup_\gamma \left|\int f\gamma \,d\mu_s\right| \leq |\mu_s|(U),$

so ${|\nu|(U)\leq |\mu_s|(U)}$. This inequality extends to Borel sets in the usual way, so ${\nu}$ is singular. $\Box$

The theorem follows relatively painlessly from the two lemmas. Fix ${\mu\in M(G)}$ with ${\hat{\mu}}$ integer-valued and let ${A = \{\gamma\mu: \int\gamma\,d\mu\neq 0\}}$. Then ${\overline{A}}$ is weak* compact, so because ${\|\cdot\|}$ is lower semicontinuous in the weak* topology there is some ${\nu\in\overline{A}}$ of minimal norm. Since ${\int d\nu}$ is an integer different from ${0}$ we must have ${\nu\neq 0}$. Thus by Lemma~2 the set ${\{\gamma\nu: \int\gamma\,d\nu\neq 0\}}$ is finite. But this implies that

$\displaystyle \nu = (n_1 \gamma_1 + \cdots + n_k \gamma_k) m_H \ \ \ \ \ (1)$

for some ${n_1,\dots,n_k\in\mathbf{Z}}$, ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and ${H=\{\gamma:\gamma\nu=\nu\}^\perp}$ the support group of ${\nu}$. In particular ${\nu}$ is absolutely continuous with respect to ${m_H}$, so because ${\nu|_H}$ is in the weak* closure of ${\{\gamma\mu|_H:\gamma\in\hat{G}\}}$ we deduce from Lemma 2 that ${\nu|_H = \gamma\mu|_H}$ for some ${\gamma}$. Thus ${\mu|_H}$ is a nonzero measure of the form (1) and we have an obvious mutually singular decomposition

$\displaystyle \mu = \mu|_H + (\mu-\mu|_H).$

Since ${\|\mu-\mu|_H\| = \|\mu\| - \|\mu|_H\|\leq\|\mu\|-1}$ the theorem follows by induction.