Abstract
Let \(W,W'\subseteq G\) be non-empty subsets in an arbitrary group G. The set \(W'\) is said to be a complement to W if \(W\cdot W'=G\) and it is minimal if no proper subset of \(W'\) is a complement to W. We show that, if W is finite then every complement of W has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal r-nets for every \(r\geqslant 0\) in finitely generated groups. Further, we give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets in finitely generated abelian groups, partially answering another problem of Nathanson. Finally, we provide infinitely many examples of infinite subsets of abelian groups of arbitrary finite rank admitting minimal complements.
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Notes
Equivalently, \({\mathscr {W}}_0\) is the inverse image of \({\mathcal {Q}}\) under the map \(\pi : {\mathscr {W}}\rightarrow {\mathbb {Z}}^d/{\mathcal {L}}\), and \({\mathscr {W}}_1\) is its complement in \({\mathscr {W}}\).
Equivalently, \({\mathbb {Z}}^d/{\mathcal {L}}\) is isomorphic to a product of copies of \({\mathbb {Z}}/2{\mathbb {Z}}\).
References
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)
Kiss, S.Z., Sándor, C., Yang, Q.-H.: On minimal additive complements of integers. J. Comb. Theory Ser. A 162, 344–353 (2019)
Nathanson, M.B.: Minimal bases and maximal nonbases in additive number theory. J. Number Theory 6, 324–333 (1974)
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)
Ruzsa, I.Z.: Additive completion of lacunary sequences. Combinatorica 21(2), 279–291 (2001). Paul Erdős and his mathematics (Budapest, 1999)
Acknowledgements
The authors would like to thank the anonymous reviewers. The first author would also like to thank the Fakultät für Mathematik, Universität Wien, where a part of the work was carried out.
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The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India.
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Biswas, A., Saha, J.P. On minimal complements in groups. Ramanujan J 55, 823–847 (2021). https://doi.org/10.1007/s11139-020-00309-3
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DOI: https://doi.org/10.1007/s11139-020-00309-3