Suppose that G is a finite Abelian group and write \({\mathcal {W}}(G)\) for the set of cosets of subgroups of G. We show that if \(f:G \rightarrow {\mathbb {Z}}\) satisfies the estimate \(\Vert f\Vert _{A(G)} \le M\) with respect to the Fourier algebra norm, then there is some \(z:{\mathcal {W}}(G) \rightarrow {\mathbb {Z}}\) such that$$\begin{aligned} f=\sum _{W \in {\mathcal {W}}(G)}{z(W)1_W}\quad \text { and }\quad \Vert z\Vert _{\ell _1({\mathcal {W}}(G))} =\exp (M^{4+o(1)}). \end{aligned}$$

中文翻译：

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科恩幂等定理的界

假设g ^是一个有限阿贝尔群和写入\（{\ mathcal {白}}（G）\）对于该组的子组的陪集ģ。我们证明，如果\（f：G \ rightarrow {\ mathbb {Z}} \）满足关于傅立叶代数范数的估计\（\ Vert f \ Vert _ {A（G）} \ le M \），那么有一些\（z：{\ mathcal {W}}（G）\ rightarrow {\ mathbb {Z}} \）使得$$ \ begin {aligned} f = \ sum _ {W \ in {\ mathcal {W}}（G）} {z（W）1_W} \ quad \ text {和} \ quad \ Vert z \ Vert _ {\ ell _1（{\ mathcal {W}}（G））} = \ exp （M ^ {4 + o（1）}）。\ end {aligned} $$