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Decomposition of the integers as a direct sum of two subsets. (English) Zbl 0824.11006

David, Sinnou (ed.), Number theory. Séminaire de théorie des nombres de Paris 1992-93. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 215, 261-276 (1995).
Two sets \(A\), \(B\) of integers induce a direct decomposition of \(\mathbb Z\) if every integer can be written uniquely as \(a+b\), \(a\in A\), \(b\in B\). While direct decompositions of \(\mathbb N\) are easily characterized, in a sense this is impossible for \(\mathbb Z\) [C. Swenson, Pac. J. Math. 53, 629–633 (1974; Zbl 0306.10031)]. The author formulates the following conjecture: for such pairs, if \(0\in A\cap B\), \(\text{gcd}_{\alpha\in A} a=1\) and \(A\) has \(n\) elements, then there is a prime \(p\mid n\) which divides all elements of \(B\). If true, this would yield an inductive characterization of direct decompositions with finite \(A\). The conjecture is established in the particular case when \(n\) is a prime-power. A certain infinite set, namely the set of finite sums of distinct odd powers of 2, is also considered.
For the entire collection see [Zbl 0814.00015].

MSC:

11B13 Additive bases, including sumsets
11B83 Special sequences and polynomials
11B34 Representation functions

Citations:

Zbl 0306.10031
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