Sets arising as minimal additive complements in the integers

Abstract

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