I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-linearity of the complexity function. Unfortunately, this was not enough to solve the question I have in mind. But, my question is the following:
Any algebraic irrational number contains at least one 0 in its expansion in basis $b$ for all $b\geq 2$ sufficiently large (or at least for $b$ in the form $10^s$, for every $s$ sufficiently large)?