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ON A PROBLEM OF NATHANSON ON NONMINIMAL ADDITIVE COMPLEMENTS

Part of: Sequences and sets

Published online by Cambridge University Press:  15 December 2025

SHI-QIANG CHEN Open the ORCID record for SHI-QIANG CHEN [Opens in a new window]  and
YUCHEN DING Open the ORCID record for YUCHEN DING [Opens in a new window]
SHI-QIANG CHEN
Affiliation:
School of Mathematics and Statistics, Anhui Normal University , PR China e-mail: csq20180327@163.com
YUCHEN DING*
Affiliation:
School of Mathematical Sciences, Yangzhou University , PR China HUN-REN Alfréd Rényi Institute of Mathematics , Budapest, Hungary
*
e-mail: ycding@yzu.edu.cn
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Abstract

Let C and W be two sets of integers. If $C+W=\mathbb {Z}$, then C is called an additive complement to W. We further call C a minimal additive complement to W if no proper subset of C is an additive complement to W. Answering a problem of Nathanson in part, we give sufficient conditions to show that W has no minimal additive complements. Our result extends a result of Chen and Yang [‘On a problem of Nathanson related to minimal additive complements’, SIAM J. Discrete Math. 26 (2012), 1532–1536].

Keywords

integer setsminimal additive complement

MSC classification

Primary: 11B13: Additive bases, including sumsets

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 8
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

The first author is supported by National Natural Science Foundation of China (Grant No. 12301003), Anhui Provincial Natural Science Foundation (Grant No. 2308085QA02) and University Natural Science Research Project of Anhui Province (Grant No. 2022AH050171). The second author is supported by National Natural Science Foundation of China (Grant No. 12201544), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20210784), China Postdoctoral Science Foundation (Grant No. 2022M710121).

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