Infinite Co-minimal Pairs in the Integers and Integral Lattices

Arindam Biswas, Jyoti Prakash Saha

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations


Given two nonempty subsets A, B of a group G, they are said to form a co-minimal pair if A· B= G, and A· B⊊ G for any ∅ ≠ A⊊ A and A· B⊊ G for any ∅ ≠ B⊊ B. The existence of co-minimal pairs is a stronger criterion than the existence of minimal complements. In this work, we show several new results about them. The existence and the construction of co-minimal pairs in the integers, with both the subsets A and B (A is not a translate of B) of infinite cardinality was unknown. We show that such pairs exist and give the first explicit construction of these pairs. The constructions also satisfy a number of algebraic properties. Further, we prove that for any d≥ 1, the group Z2 d admits infinitely many automorphisms such that for each such automorphism σ, there exists a subset A of Z2 d such that A and σ(A) form a co-minimal pair.

Original languageEnglish
Title of host publicationCombinatorial and Additive Number Theory IV, CANT 2019 and 2020
EditorsMelvyn B. Nathanson
Number of pages31
ISBN (Print)9783030679958
StatePublished - 1 Jan 2021
Externally publishedYes
EventWorkshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020 - Virtual, Online
Duration: 1 Jun 20205 Jun 2020

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceWorkshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020
CityVirtual, Online


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

ASJC Scopus subject areas

  • Mathematics (all)


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