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 | aᵐb=baⁿ. The sum of dilates, denoted as r · A + s · B for integers r and s, is defined as + sb; a∈ A, b∈ B\. In 2014, Freiman et al. {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. {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 $BS(1,3)$.