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A quantitative version of the idempotent theorem in harmonic analysis. (English) Zbl 1170.43003

Let \(G\) be a locally compact abelian group with dual group \(\widehat{G}\), and let \(\mathbf{M}(G)\) denote the measure algebra of \(G\). For \(\mu\in\mathbf{M}(G)\) we say that \(\mu\) is idempotent, if \(\mu *\mu=\mu\). The classical idempotent theorem of Paul Cohen asserts that \(\mu\) is idempotent if and only if \(\{\gamma\in\widehat{G}:\widehat{\mu}(\gamma)=1\}\) lies in the coset ring of \(\widehat{G}\), that is to say \(\widehat{\mu}=\sum_{j=1}^L\pm 1_{\gamma_j+\Gamma_j}\), where the \(\Gamma_j\) are open subgroups of \(\widehat{G}\). When \(G\) is finite this theorem gives no information, and it is the aim of the paper under review to introduce a quantitative version of it with nontrivial content for the case of finite groups. More precisely, the authors prove that:
Theorem (Quantitative idempotent theorem). Suppose that \(\mu\in\mathbf{M}(G)\) is idempotent. Then we may write \(\widehat{\mu}=\sum_{j=1}^L\pm 1_{\gamma_j+\Gamma_j}\), where \(\gamma_j\in\widehat{G}\), each \(\Gamma_j\) is an open subgroup of \(\widehat{G}\) and \(L\leq\exp(\exp(C\|\mu\|^4))\) for some absolute constant \(C\). The number of distinct subgroups of \(\Gamma_j\) may be bounded above by \(\|\mu\|+\frac1{100}\).
The norm \(\|\mu\|\) is the \(l^1\)-norm of the Fourier transform of \(f\), also known as the algebra norm \(\|f\|_A\). The following theorem is the main result of the paper and as is shown at the end of the paper in Appendix A, it gives the above quantitative version of the idempotent theorem.
Theorem (Main theorem, finite version). Suppose that \(G\) is a finite abelian group and that \(f:G\rightarrow\mathbb{Z}\) is a function with \(\|f\|_A\leq M\). Then \(f=\sum_{j=1}^L\pm 1_{x_j+H_j}\), where \(x_j\in G\), each \(H_j\leq G\) is a subgroup and \(L\leq\exp(\exp(CM^4))\). Furthermore, the number of distinct subgroups \(H_j\) may be bounded by \(M+\frac1{100}\).
The authors derive the above main theorem from a key lemma, and then prove this lemma by means of Bourgain systems. Also, they introduce a weak Freiman theorem, and finally they give some notes for possible improvements.

MSC:

43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
22B10 Structure of group algebras of LCA groups
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