  In this paper we give a generalization of the main results in \cite{ab,ab1}
about $b$-ary expansions of algebraic numbers. As a byproduct we get a large
class of new transcendence criteria. One of our corollaries implies that
$b$-ary expansions of linearly independent irrational algebraic numbers are
quite independent. Motivated by this result, we propose a generalized Borel
conjecture.
