TY - JOUR

T1 - On additive co-minimal pairs

AU - Biswas, Arindam

AU - Saha, Jyoti Prakash

N1 - Funding Information:
We wish to thank the anonymous reviewer for the constructive comments and suggestions. The first author would like to acknowledge the support of the OWLF program of the Mathematisches Forschungsinstitut Oberwolfach (MFO) and would also like to thank the Fakultät für Mathematik, Universität Wien. The second author would like to acknowledge the Initiation Grant IISERB/R&D/2018-19/224 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.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/6/1

Y1 - 2021/6/1

N2 - A pair of non-empty subsets (W,W′) in an abelian group G is an additive complement pair if W+W′=G. The set W′ is said to be minimal to W if W+(W′∖{w′})≠G,∀w′∈W′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs (W,W′) such that both W and W′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of G.

AB - A pair of non-empty subsets (W,W′) in an abelian group G is an additive complement pair if W+W′=G. The set W′ is said to be minimal to W if W+(W′∖{w′})≠G,∀w′∈W′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs (W,W′) such that both W and W′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of G.

KW - Additive complements

KW - Additive number theory

KW - Minimal complements

KW - Representation of integers

KW - Sumsets

UR - http://www.scopus.com/inward/record.url?scp=85099392820&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2020.10.010

DO - 10.1016/j.jnt.2020.10.010

M3 - Article

AN - SCOPUS:85099392820

VL - 223

SP - 350

EP - 370

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -