Let A and B be sets of integers. For integers r and s, the sum of dilates, denoted by \(r \cdot A + s \cdot B,\) is defined as \(\{ra + sb; a\in A, b\in B\}\) . For integers m and n, the Baumslag-Solitar group, denoted by BS(m, n), is the group generated by two elements with a single defining relation: \(BS(m,n) = \langle a, b | a^mb=ba^n\rangle \) . In 2014, Freiman et al. [10] derived direct and inverse results for sums of dilates and applied these in order to address specific direct and inverse problems within the Baumslag-Solitar group, assuming suitable small doubling properties. In 2015, Freiman et al. [11] proved the general problem of small doubling types for a subset of the Baumslag-Solitar group BS(1, 2). In this paper, we extend these investigations to solve the analogous problem for the Baumslag-Solitar group BS(1, 3).
