Hereditarily nonparadoxical sets revisited

We prove that assuming c<ℵω every translation - hereditarily nonparadoxical subset of Rl 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].