On the independence of greedy expansions of certain algebraic numbers in a Pisot or Salem base

Let β be a Pisot or Salem number with β > 1, and let α₁ and α₂ be elements in \({\mathbb{Q}}\) $\mathbb{Q}$ (β) ∩ [0, 1). In this note, we prove that α₁ and α₂ have either the same tail greedy expansions in a base β or independent random greedy expansions in a base β.