On non-minimal complements

Arindam Biswas, Jyoti Prakash Saha

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Article number102226
JournalAdvances in Applied Mathematics
Volume130
DOIs
StatePublished - 1 Sep 2021
Externally publishedYes

Keywords

  • Additive complements
  • Additive number theory
  • Minimal complements
  • Representation of integers
  • Sumsets

ASJC Scopus subject areas

  • Applied Mathematics

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