On the set of points represented by harmonic subseries

We show that the set of points \[ (∑n∈ A1/n, ∑n∈ A1/n+1, ∑n∈ A1/n+2), \] obtained as A ranges over infinite sets of positive integers, has a non-empty interior. This answers in the affirmative a question of Erd{o}s and Straus (posed in a 1980 problem book by Erd{o}s and Graham).