  Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we
denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots
a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$.
The sets $\mathscr U_k$ are one of the most fundamental invariants studied in
the theory of non-unique factorization, and understanding their structure is a
basic problem in the field: In particular, it is known that, in many cases of
interest, these sets are almost arithmetic progressions with the same
difference and bound for all large $k$, namely, $H$ satisfies the Structure
Theorem for Unions. The present paper improves the current state of the art on
this problem.
  More precisely, we show that, under mild assumptions on $H$, not only does
the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$
such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr
U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad
\bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are
$\mu$-periodic from some point on. The result applies, e.g., to (the
multiplicative monoid of) all commutative Krull domains (e.g., Dedekind
domains) with finite class group; a variety of weakly Krull commutative domains
(including all orders in number fields with finite elasticity); some maximal
orders in central simple algebras over global fields; and all numerical
monoids.
  Large parts of the proofs are worked out in a "purely additive model", by
inquiring into the properties of what we call a subadditive family, i.e., a
collection $\mathscr L$ of subsets of $\bf N$ such that, for all
$L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.
