The realization problem for delta sets of numerical semigroups

Abstract

The delta set of a numerical semigroup $S$, denoted $\Delta (S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in~$S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta (S) = \gcd \Delta (S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup~$S$ with this delta set. We then show that, for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension~3 if and only if $t=2$.