超字符、指数和与测不准原理
Supercharacters, exponential sums, and the uncertainty principle
摘要：
超特征标理论是最近由P.Diaconis和I.M. Isaacs，在C.安德鲁我们研究了由某些矩阵群的作用引起的$(Z/nZ)^d$上的超特征标理论，证明了数论中感兴趣的各种指数和（例如，Gauss、Ramanujan、Heilbronn和Kloosterman和）以这种方式出现。我们开发了一个推广的离散傅里叶变换，其中超字符的傅里叶指数基础的作用。我们提供了一个相应的不确定性原理，并计算了几种情况下的相关常数。
英文摘要：
The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.
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