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

Abstract

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 C ₁ N^{σ (β)}, where and C ₁ is a constant depending only on β, if all prime ideal factors of β are unramified and their norms are integer primes.