Is the countability of the set of irrational algebraic numbers somehow reflected in a characteristic property of their decimal expansions? 



Question on the decimal expansion of algebraic numbers
The answer is not known, and it is a conjecture of Borel that the answer is no. See [Words and Transcendence](http://arxiv.org/abs/0908.4034) by M. Waldschmidt for references, in particular your previous claim that every digit in irrational algebraic numbers occurs infinitely often is an open problem. We don't know if there are irrational algebraic numbers in the Cantor set.


