Inverse problems for minimal complements and maximal supplements

2020/05/31 by Alon, Noga, Kravitz, Noah, Larson, Matt
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.00534

Abstract

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 |C| ≤ 2^2/3|G|^1/3/((3e log |G|)^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.

Related

• Sets Arising as Minimal Additive Complements in the Integers
• Asymptotic behaviour of minimal complements
• On a problem of Nathanson on non-minimal additive complements
• On minimal additive complements of integers
• On Eventually Periodic Sets as Minimal Additive Complements