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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Minimal complementary sets


Author: Gerald Weinstein
Journal: Trans. Amer. Math. Soc. 212 (1975), 131-137
MSC: Primary 10J20
MathSciNet review: 0399023
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Abstract: Let G be a group on which a measure m is defined. If $ A,B \subset G$ we define $ A \oplus B = C = \{ c\vert c = a + b,a \in A,b \in B\} $. By $ {A_k} \subset G$ we denote a subset of G consisting of k elements. Given $ {A_k}$ we define $ s({A_k}) = \inf m\{ B\vert B \subset G,{A_k} \oplus B = G\} $ and $ {c_k} = {\sup _{{A_k} \subset G}}s({A_k})$. Theorems 1, 2, and 3 deal with the problem of determining $ {c_k}$.

In the dual problem we are given B, $ m(B) > 0$, and required to find minimal A such that $ A \oplus B = G$ or, sometimes, $ m(A \oplus B) = m(G)$. Theorems 5 and 6 deal with this problem.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0399023-0
PII: S 0002-9947(1975)0399023-0
Article copyright: © Copyright 1975 American Mathematical Society