Hereditarily nonparadoxical sets revisited

We prove that assuming c < ℵ ω every translation – hereditarily nonparadoxical subset of R l is a countable sum of sets without repeated differences. In particular, under the same assumption every hereditarily nonparadoxical subset of the real line is a countable sum of sets without repeated distances. This gives a partial answer to the question of M. Penconek from [4].