1. Minimal additive complements in finitely generated abelian groups
- Author
-
Arindam Biswas and Jyoti Prakash Saha
- Subjects
Rank (linear algebra) ,Sumsets ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,Minimal complements ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,FOS: Mathematics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Finitely-generated abelian group ,0101 mathematics ,Abelian group ,Additive complements ,Complement (set theory) ,Mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Additive number theory ,010102 general mathematics ,Number theory ,010201 computation theory & mathematics ,11B13, 05E15, 05B10 ,Combinatorics (math.CO) ,Mathematics - Group Theory - Abstract
Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ 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 work by him, Chen--Yang, Kiss--S\`andor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this article, we study subsets which are not eventually periodic. We introduce the notion of "spiked subsets" and give necessary and sufficient conditions for the existence of minimal complements for them. This provides a partial answer to a problem of Nathanson., Comment: 25 pages, 8 figures
- Published
- 2021