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 -