Sets arising as minimal additive complements in the integers

Abstract: A subset $C$ of a group $G$ is a \emph{minimal additive complement} to $W \subseteq G$ if $C +W = G$ and if $C' + W \neq G$ for any proper subset $C'\subsetneq C$. Work started by Nathanson has focused on which sets $W\subseteq \mathbb{Z}$ have minimal additive complements. We instead investigate which sets $C\subseteq \mathbb{Z}$ arise as minimal additive complements to some set $W\subseteq \mathbb{Z}$. We confirm a conjecture of Kwon in showing that bounded below sets containing arbitrarily large gaps arise as minimal additive complements. We provide partial results for determining which eventually periodic sets arise as minimal additive complements. We place bounds on the density of sets which arise as minimal additive complements to finite sets, including periodic sets which arise as minimal additive complements. We conclude with several conjectures and questions concerning the structure of minimal additive complements.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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