On minimal additive complements of integers

Abstract

Let C, W ⊆ Z. If C + W = Z, then the set C is called an additive complement to W in Z. If no proper subset of C is an additive complement to W , then C is called a minimal additive complement. Let X ⊆ N. If there exists a positive integer T such that x+T ∈ X for all sufficiently large integers x ∈ X, then we call X eventually periodic.In this paper, we study the existence of a minimal complement to W when W is eventually periodic or not. This partially answers a problem of Nathanson.

A note on minimal additive complements of integers

“…These results together indicate that in the case of sets of this special form, one can directly identify a "source" of irregularity that permits a minimal additive complement to exist, namely the subset G. As an added consequence of these results, we can easily deduce the following theorem from [2] without the explicit construction that was used.…”

“…Theorem 8 (Kiss, Sándor, Yang [2]). There exists an infinite, not eventually periodic set W ⊆ N such that wi+1 − wi ∈ {1, 2} for all i and there exists a minimal complement to W .…”

“…Thus, if G is finite this result roughly allows us to reduce the existence of a minimal additive complement of W in Z to the existence of a minimal additive complement to W mod n in Z/nZ. In [2], Kiss, Sándor and Yang pose the question of whether Theorem 5 is still true when G is infinite. We observe that it is certainly not true in general, for if G is of the form nN + B, then W = nN + (A ∪ B), and such a set does not have a minimal additive complement.…”

“…In [2], Kiss, Sándor and Yang consider a nice collection of such sets, namely those W ⊆ Z that are bounded below and for which there exists a positive integer n such that w + n ∈ W for sufficiently large w ∈ W , which they call eventually periodic sets. Such sets, after some point, consist of a union of arithmetic progressions modulo n; the elements prior to that can be distinguished based on whether or not they are in the same residue classes modulo n as the eventual arithmetic progressions.…”

Inverse problems for minimal complements and maximal supplements

“…It is easy to check that, for instance, N does not have a minimal complement in Z. Kiss, Sándor, and Yang [10] studied the existence of minimal complements for "eventually periodic" subsets of Z. There has been some progress in infinite abelian groups other than Z: Biswas and Saha [5] generalized Nathanson's result by showing that if G is any abelian group and W is a finite non-empty subset of G, then any complement to W contains a minimal complement.…”