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 | a^mb=ba^n⟩. The sum of dilates, denoted as r · A + s · B for integers r and s, is defined as {ra + sb; a∈ A, b∈ B}. In 2014, Freiman et al. 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. 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).