A small doubling structure theorem in a Baumslag-Solitar group

Let G denote an arbitrary group. If X is a subset of G, we define its square X^2 by X^2 = {ab | a, b ∈ X}. This paper deals with the following type of problems. Let X be a finite subset of a group G. Determine the structure of X if the following inequality holds: |X^2| ≤ α|X| + β for some small α ≥ 1 and small |β|. Such problems are called inverse problems of small doubling type. We solve a general inverse problem of small doubling type in a monoid, which is a subset of the Baumslag–Solitar group BS(1, 2). Here the Baumslag-Solitar groups BS(m, n) are two-generated groups with one relation, which are defined as follows: BS(m, n) = , where m and n are integers.