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

Abstract

Let b ⩾ 2 be an integer. According to a conjecture of Émile 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 α and α′ 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’.