Non-Wieferich property of prime ideals and a conjecture of Erdös

Let $K$ be a number field with ring of integers $\mathcal{O}$ and $α\in\mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $α$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $β\in\mathcal{O}$ is relatively prime to $α$ and all prime ideal factors of $(β)$ are unramified and have residue degree $1$, we apply our higher $α$-Wieferich property to establish the asymptotic equidistribution of digits in $β$-adic expansions of $α^n$, which is a generalization of the Dupuy-Weirich theorem. When $(β)$ have ramified prime ideal factors, we also obtain a result on the block complexity of $β$-adic expansions of $α^n$.