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 $\widehat{G}$.
@article{10_4007_annals_2008_168_1025,
author = {Ben Green and Tom Sanders},
title = {A quantitative version of the idempotent theorem in harmonic analysis},
journal = {Annals of mathematics},
pages = {1025--1054},
year = {2008},
volume = {168},
number = {3},
doi = {10.4007/annals.2008.168.1025},
mrnumber = {2456890},
zbl = {1170.43003},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.1025/}
}
TY - JOUR AU - Ben Green AU - Tom Sanders TI - A quantitative version of the idempotent theorem in harmonic analysis JO - Annals of mathematics PY - 2008 SP - 1025 EP - 1054 VL - 168 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.1025/ DO - 10.4007/annals.2008.168.1025 LA - en ID - 10_4007_annals_2008_168_1025 ER -
%0 Journal Article %A Ben Green %A Tom Sanders %T A quantitative version of the idempotent theorem in harmonic analysis %J Annals of mathematics %D 2008 %P 1025-1054 %V 168 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.1025/ %R 10.4007/annals.2008.168.1025 %G en %F 10_4007_annals_2008_168_1025
Ben Green; Tom Sanders. A quantitative version of the idempotent theorem in harmonic analysis. Annals of mathematics, Tome 168 (2008) no. 3, pp. 1025-1054. doi: 10.4007/annals.2008.168.1025
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