Abstract
Given two nonempty subsets \(W,W'\subseteq G\) in an arbitrary abelian group G, the set \(W'\) is said to be an additive complement to W if \(W + W'=G\) and it is minimal if no proper subset of \(W'\) is a complement to W. The notion was introduced by Nathanson and previous works by him, Chen–Yang, Kiss–Sándor–Yang, etc. focussed on \(G =\mathbb {Z}\). In this article, we focus on the higher rank case. We introduce the notion of “spiked subsets” and give necessary and sufficient conditions for the existence of minimal complements for them. This provides an answer to a problem of Nathanson in several contexts.
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Notes
\(\mathbb {Z}^r\) is considered as a subgroup of \(\mathbb {Z}^d\) defined by vanishing of \(d-r\) coordinates.
\(\tan \frac{\pi }{2}\) is to be interpreted as “\(\frac{1}{0}\)” and hence as a rational.
\(\tan \frac{\pi }{2}\) is to be interpreted as “\(\frac{1}{0}\)” and hence as a rational.
References
Biswas, A., Saha, J.P.: On minimal complements in groups. Ramanujan J. (2020). https://doi.org/10.1007/s11139-020-00309-3
Chen, Y.-G., Yang, Q.-H.: On a problem of Nathanson related to minimal additive complements. SIAM J. Discret. Math. 26(4), 1532–1536 (2012)
Erdős, P.: Some results on additive number theory. Proc. Am. Math. Soc. 5, 847–853 (1954)
Erdős, P.: Some unsolved problems. Mich. Math. J. 4, 291–300 (1957)
Kiss, S.Z., Sándor, C., Yang, Q.H.: On minimal additive complements of integers. J. Combin. Theory Ser. A 162, 344–353 (2019)
Lorentz, G.G.: On a problem of additive number theory. Proc. Am. Math. Soc. 5, 838–841 (1954)
Nathanson, M.B.: Problems in additive number theory, IV: Nets in groups and shortest length \(g\)-adic representations. Int. J. Number Theory 7(8), 1999–2017 (2011)
Acknowledgements
We wish to thank the anonymous reviewer. The first author acknowledges the fellowship of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) and would also like to thank the Fakultät für Mathematik, Universität Wien where a part of the work was carried out.
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Jyoti Prakash would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award IFA18-MA123 from the Department of Science and Technology, Government of India.
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Biswas, A., Saha, J.P. Minimal additive complements in finitely generated abelian groups. Ramanujan J 57, 215–238 (2022). https://doi.org/10.1007/s11139-021-00421-y
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DOI: https://doi.org/10.1007/s11139-021-00421-y