调和分析中幂等定理的一个定量形式
A quantitative version of the idempotent theorem in harmonic analysis
摘要：
设G是局部紧阿贝尔群，记M（G）为卷积下有界正则复值测度代数。M（G）中的测度μ称为幂等的，如果μ * μ = μ，或者如果傅里叶-斯蒂尔切斯变换μ ^仅取0和1。Cohen-Helson-Rudin幂等定理指出测度μ是幂等的当且仅当集合{r in G ^：μ ^（r）= 1}属于G ^的陪集环，也就是说我们可以将μ ^写成陪集r_j + H_j的特征函数的有限加/减1组合，其中H_j是G ^的开子群。本文证明了这种陪集的个数L可以按范数有界||μ||，并且实际上可以取L <= \exp \exp（C|| μ|| ^4）。特别地，我们的结果是非平凡的，甚至对有限群。
英文摘要：
Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure μin M(G) is said to be idempotent if μ* μ= μ, or alternatively if the Fourier-Stieltjes transform μ^ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure μis idempotent if and only if the set {r in G^ : μ^(r) = 1} belongs to the coset ring of G^, that is to say we may write μ^ as a finite plus/minus 1 combination of characteristic functions of cosets r_j + H_j, where the H_j are open subgroups of G^. In this paper we show that the number L of such cosets can be bounded in terms of the norm ||μ||, and in fact one may take L <= \exp\exp(C||μ||^4). In particular our result is non-trivial even for finite groups.
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