I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture).
I know that by using Ridout theorem or Schmidt subspace theorem is possible to find a good lower bound for the number of nonzero "digits" in the $g$-ary expansion of an algebraic irrational number (for any basis $g\geq 2$).
However, my question is in the opposite direction:
Is it possible to prove that every algebraic irrational number has at least one 0 in its $g$-ary expansion, for all sufficiently large $g\geq 2$?
Of course, if this statement is true, then it is possible to prove that, in fact, there are infinitely many $0$'s in its $g$-ary expansion (by multiplying the algebraic number for some convenient power of $10$).
Any help will be welcomed.