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

 

Sidon sets with extremal Sidon constants


Authors: Colin C. Graham and L. Thomas Ramsey
Journal: Proc. Amer. Math. Soc. 83 (1981), 522-526
MSC: Primary 43A46; Secondary 20F99
MathSciNet review: 627683
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Abstract: A finitely supported measure $ \mu $ on an l.c.a. group is said to be extremal if $ {\left\Vert {\hat \mu } \right\Vert _\infty } = {\left\Vert \mu \right\Vert^{1/2}} = {(\char93 {\text{supp}}\;\mu {\text{)}}^{1/2}}$. If $ \mu $ is an extremal measure and $ E$ is the support of $ \mu $, it follows that the Sidon constant of $ E$ is $ {(\char93 E)^{1/2}}$, in which case $ E$ is also said to be extremal. Our results are these. (1) An "independent" union of $ m$ cosets of a finite subgroup $ H$ of $ G$ is extremal if and only if (essentially) $ m$ divides $ \char93 H$. (2) Not all extremal subsets of abelian groups have the form described in (1). (3) For any group (abelian or not), the Sidon constant of that group is at least $ (.8){(\char93 G)^{1/13}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0627683-3
PII: S 0002-9939(1981)0627683-3
Article copyright: © Copyright 1981 American Mathematical Society