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 | References | Similar Articles | Additional Information

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.

- 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