On β-adic expansions of powers of algebraic integer omitting a digit

Let α, β be two relatively prime algebraic integers in a number field K and N be a positive integer. We show that the number of n∈\1,2,,N\ such that the β-adic expansion of αⁿ omits a given digit is less than C₁ Nσ(β), where σ(β):=log(|N(β)|-1)/log|N(β)| and C₁ is an absolute constant, if all prime ideal factors of β are unramified and their norms are integer primes.