Inverse problems for minimal complements and maximal supplements

JOURNAL OF NUMBER THEORY

卷:223

Article

Alon, Noga^1,2,3  Kravitz, Noah⁴  Larson, Matt⁵ 

[1] Princeton Univ, Dept Math, Princeton, NJ 08544 USA

[2] Tel Aviv Univ, Sch Math, IL-69978 Tel Aviv, Israel

[3] Tel Aviv Univ, Sch Comp Sci, IL-69978 Tel Aviv, Israel

[4] Zoom Univ Yale, Grace Hopper Coll, New Haven, CT 06511 USA

[5] Dept Math, 450 Jane Stanford Way, Stanford, CA 94305 USA

关键词: Minimal complement;    Additive combinatorics;    Probabilistic combinatorics;   

DOI  :  10.1016/j.jnt.2020.10.009

来源: Elsevier

【 摘 要 】

Given a subset W of an abelian group G, a subset C is called an additive complement for W if W + C = G; if, moreover, no proper subset of C has this property, then we say that C is a minimal complement for W. It is natural to ask which subsets C can arise as minimal complements for some W. We show that in a finite abelian group G, every non-empty subset C of size vertical bar C vertical bar <= 2(2)/(3)vertical bar G vertical bar(1/3)/((3e log vertical bar G vertical bar)(2/3) is a minimal complement for some W. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for dual problems about maximal supplements. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free