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 \ra + 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).