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Complementing sets of $n$-tuples of integers


Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 34 (1972), 71-72
MSC: Primary 10L05
DOI: https://doi.org/10.1090/S0002-9939-1972-0294286-7
MathSciNet review: 0294286
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Abstract: Let S, ${A_1},{A_2}, \cdots ,{A_p}$ be finite nonempty sets of n-tuples of integers such that, if ${a_i} \in {A_i}$, for $i = 1,2, \cdots ,p$, then ${a_1} + {a_2} + \cdots + {a_p} \in S$, and such that every $s \in S$ has a unique representation as a sum $s = {a_1} + {a_2} + \cdots + {a_p}$ with ${a_i} \in {A_i}$. If S is the cartesian product of n sets of integers, then each ${A_i}$ is also the cartesian product of n sets of integers, and conversely.


References [Enhancements On Off] (What's this?)

  • Rodney T. Hansen, Complementing pairs of subsets of the plane, Duke Math. J. 36 (1969), 441–449. MR 244404
  • Ivan Niven, A characterization of complementing sets of pairs of integers, Duke Math. J. 38 (1971), 193–203. MR 274414

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Keywords: Complementing sets, sumsets of integers, addition of <I>n</I>-tuples of integers
Article copyright: © Copyright 1972 American Mathematical Society