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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Representation functions of sequences in additive number theory

Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 72 (1978), 16-20
MSC: Primary 10L05
MathSciNet review: 503522
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Abstract: Let $ \mathcal{A}$ be a set of nonnegative integers, and let $ r_2^\mathcal{A}(n)$ denote the number of representations of n in the form $ n = {a_i} + {a_j}$ with $ {a_i},{a_j} \in \mathcal{A}$. The set $ \mathcal{A}$ is periodic if $ a \in \mathcal{A}$ implies $ a + m \in \mathcal{A}$ for some $ m \geqslant 1$ and all $ a > N$. It is proved that if $ \mathcal{A}$ is not periodic, then for every set $ \mathcal{B} \ne \mathcal{A}$ there exist infinitely many n such that $ r_2^\mathcal{A}(n) \ne r_2^\mathcal{B}(n)$. Moreover, all pairs of periodic sets $ \mathcal{A}$ and $ \mathcal{B}$ are constructed that satisfy $ r_2^\mathcal{A}(n) = r_2^\mathcal{B}(n)$ for all but finitely many n.

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Keywords: Representation functions, addition of sequences, sum sets
Article copyright: © Copyright 1978 American Mathematical Society