TY - GEN

T1 - Infinite Co-minimal Pairs in the Integers and Integral Lattices

AU - Biswas, Arindam

AU - Saha, Jyoti Prakash

N1 - Funding Information:
Acknowledgements We thank the anonymous referee for a careful reading of the manuscript and for a number of helpful comments and suggestions. The first author is supported by the ISF Grant no. 662/15 and the ISF Grant no. 1226/19. He would also like to thank the Department of Mathematics at the Technion where a part of the work was carried out. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India.
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 -