Direct and Inverse Problems in Baumslag-Solitar Group $BS(1,3)$

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: $BS(m,n)=⟨a,b\mathrm{\mid }{a}^{m}b=b{a}^n⟩$. 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 $BS(1,3)$.