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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Extreme values for the Sidon constant

Authors: Donald I. Cartwright, Robert B. Howlett and John R. McMullen
Journal: Proc. Amer. Math. Soc. 81 (1981), 531-537
MSC: Primary 43A46; Secondary 43A65
MathSciNet review: 601723
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Abstract: Let $ G$ be a compact group and let $ \phi \ne P \subseteq \hat G$. We consider the inequalities $ 1 \leqslant \kappa (P) \leqslant {({\Sigma _{\sigma \in P}}d_\sigma ^2)^{1/2}}$, where $ \kappa (P)$ denotes the Sidon constant of $ P$. The condition $ \kappa (P) = 1$ essentially characterizes an example of Figà-Talamanca and Rider. The condition $ \kappa (P) = {({\Sigma _{\sigma \in P}}d_\sigma ^2)^{1/2}}$ for finite $ P$ is equivalent to the existence of certain interesting functions on $ G$. We show that $ \kappa (\hat G) = {\left\vert G \right\vert^{1/2}}$ for a very large class of finite groups $ G$, and this implies the existence of "$ G$-circulant" unitary matrices whose entries all have modulus $ {\left\vert G \right\vert^{ - 1/2}}$.

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PII: S 0002-9939(1981)0601723-X
Keywords: Sidon constant, Sidon set, circulant Hadamard matrix
Article copyright: © Copyright 1981 American Mathematical Society