# A quantitative version of the idempotent theorem in harmonic analysis

### Abstract

Suppose that $G$ is a locally compact abelian group, and write $\mathbf{M}(G)$ for the algebra of bounded, regular, complex-valued measures under convolution. A measure $\mu \in \mathbf{M}(G)$ is said to be idempotent if $\mu \ast \mu = \mu$, or alternatively if $\widehat{\mu}$ takes only the values $0$ and $1$. The Cohen-Helson-Rudin idempotent theorem states that a measure $\mu$ is idempotent if and only if the set $\{\gamma \in \widehat{G} : \widehat{\mu}(\gamma) = 1\}$ belongs to the coset ring of $\widehat{G}$, that is to say we may write $\widehat{\mu} = \sum_{j = 1}^L \pm 1_{\gamma_j + \Gamma_j}$ where the $\Gamma_j$ are open subgroups of $\widehat{G}$.

In this paper we show that $L$ can be bounded in terms of the norm $\Vert \mu \Vert$, and in fact one may take $L \leq \exp\exp(C \Vert \mu \Vert^4)$. In particular our result is nontrivial even for finite groups.

## Authors

Ben Green

Center for Mathematical Sciences
University of Cambridge
Cambridge CB3 0WA
United Kingdom

Tom Sanders

Center for Mathematical Sciences
University of Cambridge
Cambridge CB3 0WA
United Kingdom