Characterization of complementing pairs of $(\mathbb Z_{{\geq}{0}})^{n∗}$

Let $A,B,C$ be subsets of an abelian group $G$. A pair $(A,B)$ is called a $C$-pair if $A,B \subset C$ and $C$ is the direct sum of $A$ and $B$. The $(\mathbb Z_{{\geq}{0}})$-pairs are characterized by de Bruijn in 1950 and the $(\mathbb Z_{{\geq}{0}})^2$-pairs are characterized by Niven in 1971. In this paper, we characterize the $(\mathbb Z_{{\geq}{0}})^n$-pairs for all $n \geq 1$. We show that every $(\mathbb Z_{{\geq}{0}})^n$-pair is characterized by a weighted tree if it is primitive, that is, it is not a Cartesian product of a $(\mathbb Z_{{\geq}{0}})^p$-pair and a $(\mathbb Z_{{\geq}{0}})^q$-pair of lower dimensions.

Keywords

complementing pair, power series, weighted tree

2010 Mathematics Subject Classification

11P83, 52C22

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Dedicated to the memory of Professor Ka-Sing Lau

The work is supported by NSFC Nos. 12071167 and 11971195.NSFC Nos. 12071167 and 11971195.

Received 2 June 2022

Accepted 13 September 2023

Published 7 August 2024