TY - GEN
T1 - Infinite Co-minimal Pairs in the Integers and Integral Lattices
AU - Biswas, Arindam
AU - Saha, Jyoti Prakash
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - 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.
AB - 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.
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=85108854117&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-67996-5_4
DO - 10.1007/978-3-030-67996-5_4
M3 - Conference contribution
AN - SCOPUS:85108854117
SN - 9783030679958
T3 - Springer Proceedings in Mathematics and Statistics
SP - 41
EP - 71
BT - Combinatorial and Additive Number Theory IV, CANT 2019 and 2020
A2 - Nathanson, Melvyn B.
PB - Springer
T2 - Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020
Y2 - 1 June 2020 through 5 June 2020
ER -