On the independence of expansions of algebraic numbers in an integer base

Let b >= 2 be an integer. According to a conjecture of Emile Borel, the b-adic expansion of any irrational algebraic number behaves in some respects 'like a random sequence'. We give a contribution to the following related problem: let alpha and alpha' be irrational algebraic numbers, then prove that their b-adic expansions either have the same tail, or behave in some respects 'like independent random sequences'.