TY - JOUR
T1 - On non-minimal complements
AU - Biswas, Arindam
AU - Saha, Jyoti Prakash
N1 - Funding Information:
The authors would like to thank the anonymous reviewer. The work of the first author was supported by the ISF Grant no. 662/15 . He wishes to thank the Department of Mathematics at the Technion where a part of the work was carried out. The second author acknowledges 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.
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/9/1
Y1 - 2021/9/1
N2 - The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon–Kravitz–Larson, Burcroff–Luntzlara and also that of the authors, shed light on some of the questions in this direction. These works have focussed mainly on the group of integers, or on abelian groups. In this work, our motivation is two-fold: (1) to show some new results on the inverse problem, (2) to concentrate on the inverse problem in not necessarily abelian groups. As a by-product, we obtain new results on non-minimal complements in the group of integers and more generally, in any finitely generated abelian group of positive rank and in any free abelian group of positive rank. Moreover, we show the existence of uncountably many subsets in such groups which are “robust” non-minimal complements.
AB - The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon–Kravitz–Larson, Burcroff–Luntzlara and also that of the authors, shed light on some of the questions in this direction. These works have focussed mainly on the group of integers, or on abelian groups. In this work, our motivation is two-fold: (1) to show some new results on the inverse problem, (2) to concentrate on the inverse problem in not necessarily abelian groups. As a by-product, we obtain new results on non-minimal complements in the group of integers and more generally, in any finitely generated abelian group of positive rank and in any free abelian group of positive rank. Moreover, we show the existence of uncountably many subsets in such groups which are “robust” non-minimal complements.
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=85107556703&partnerID=8YFLogxK
U2 - 10.1016/j.aam.2021.102226
DO - 10.1016/j.aam.2021.102226
M3 - Article
AN - SCOPUS:85107556703
SN - 0196-8858
VL - 130
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
M1 - 102226
ER -