On $\beta$-adic expansions of powers of algebraic integer omitting a digit

Abstract

Let $\alpha ,\beta$ be two relatively prime algebraic integers in a number field $K$ and $N$ be a positive integer. We show that the number of $n\in \{1,2,\dots ,N\}$ such that the $\beta$-adic expansion of ${\alpha }^n$ omits a given digit is less than $C_1{N}^{\sigma (\beta )}$, where $\sigma (\beta ):=\frac{\log (|N(\beta )|-1)}{\log |N(\beta )|}$ and $C_1$ is an absolute constant, if all prime ideal factors of $\beta$ are unramified and their norms are integer primes.