  For integers $m$ and $n$, the Baumslag-Solitar groups, denoted as $BS(m,n)$,
are groups generated by two elements with a single defining relation: $BS(m,n)
= \langle a, b | a^mb=ba^n\rangle$. The sum of dilates, denoted as $r \cdot A +
s \cdot B$ for integers $r$ and $s$, is defined as $\{ra + sb; a\in A, b\in
B\}$. In 2014, Freiman et al. \cite{freiman} derived direct and inverse results
for sums of dilates and applied these findings to address specific direct and
inverse problems within Baumslag-Solitar groups, assuming suitable small
doubling properties. In 2015, Freiman et al. \cite{freiman15} tackled the
general problem of small doubling types in a monoid, a subset of the
Baumslag-Solitar group $BS(1,2)$. This paper extends these investigations to
solve the analogous problem for the Baumslag-Solitar group $BS(1,3)$.
