Union of sets of lengths of numerical semigroups

Abstract: Let $S=\langle a_1,\ldots,a_p\rangle$ be a numerical semigroup, $s\in S$ and ${\sf z}(s)$ its set of factorizations. The set of length is denoted by ${\mathcal L}(s)={{\tt L}(x_1,\dots,x_p)\mid (x_1,\dots,x_p)\in{\sf Z}(s)}$ where ${\tt L}(x_1,\dots,x_p)=x_1+\ldots+x_p$. From these definitions, the following sets can be defined ${\textsf W}(n)={s\in S\mid \exists x\in{\sf z}(s) \textrm{ such that } {\tt{L}}(x)=n}$, $\nu(n)=\cup_{s\in {\textsf W}(n)} {\mathcal L}(s)={l_1<l_2<\ldots< l_r}$ and $\Delta\nu(n)={l_2-l_1,\ldots,l_r-l_{r-1}}$. In this paper, we prove that the set $\Delta\nu(S)=\cup_{n\in{\mathbb{N}}}\Delta\nu(n)$ is almost periodic with period ${\rm lcm}(a_1,a_p)$.