Sets arising as minimal additive complements in the integers

A subset C of an abelian group G is said to be a minimal additive complement W⊆G if C+W=G and C′+W≠G for any proper C′⊂C. In this paper, we study certain subsets the integers that arise or do not as complements. Following suggestion Kwon, show bounded-below sets with arbitrarily large gaps Moreover, our construction shows such set belongs co-minimal pair, strengthening result Biswas Saha lacunary sequences in integers. We bound upper lower Banach density syndetic complements finite sets. provide some necessary conditions eventually periodic demonstrate these are also sufficient classes conclude several conjectures questions concerning structure