  The aim of this paper is threefold:
  a) Finding new direct and inverse results in the additive number theory
concerning Minkowski sums of dilates.
  b) Finding a connection between the above results and some direct and inverse
problems in the theory of Baumslag-Solitar (non-abelian) groups.
  c) Solving certain inverse problems in Baumslag-Solitar groups or monoids,
assuming appropriate small doubling properties.
