Maximally additively reducible subsets of the integers

Abstract

Let $A, B \subseteq \mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \subseteq \mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\{0, 1, \ldots, k\}$ which have $0$ as an element, the full interval $\{0, 1, \ldots,k\}$ has the most divisors. To generalize to sets which do not have $0$ as an element, we prove a correspondence between additive divisors and lunar multiplication, introduced by Appelgate, LeBrun and Sloane (2011) in their study of a kind of min/max arithmetic. The number of binary lunar divisors is related to compositions of integers which are restricted in that the first part is greater or equal to all other parts. We establish some bounds on such compositions to show that $\{1, \ldots, k\}$ has the most divisors among all subsets of $\{0, 1, \ldots, k\}$. These results resolve two conjectures of LeBrun et al. regarding the maximal number of lunar binary divisors, a special case of a more general conjecture about lunar divisors in arbitrary bases. We resolve this third conjecture by generalizing from sum-sets to sum-multisets.

Figures (5)

• Figure 1: Example of $k$-promotion
• Figure 2: Example of $F(B)$ for $A = \{0, 2, 3, 4, 5, 6\}$
• Figure 3: Lunar arithmetic
• Figure 4: Two representations of sumsets
• Figure 5: Two representations of multiset addition

Theorems & Definitions (57)

• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof
• Lemma 7