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For sufficiently nice families of semigroups and monoids, the structuretheorem for sets of length states that the length set of any sufficiently largeelement is an arithmetic sequence with some values omitted near the ends. Inthis paper, we prove a specialized version of the structure theorem that holdsfor any numerical semigroup $S$. Our description utilizes two other numericalsemigroups $S_{\mathsf M}$ and $S_{\mathsf m}$, derived from the generators of$S$: for sufficiently large $n \in S$, the Ap\'ery sets of $S_{\mathsf M}$ and$S_{\mathsf m}$ specify precisely which lengths appear in the length set of$n$, and their gaps specify which lengths are "missing". We also provide anexplicit bound on which elements satisfy the structure theorem.

The structure theorem for sets of length for numerical semigroups