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arXiv · 2405.06220

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

Jiuzhou Zhao·Ruofan Li
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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 $n\in\{1,2,\dots,N\}$ such that the $β$-adic expansion of $α^n$ omits a given digit is less than $C_1 N^{σ(β)}$, where $σ(β):=\frac{\log(|N(β)|-1)}{\log|N(β)|}$ and $C_1$ is an absolute constant, if all prime ideal factors of $β$ are unramified and their norms are integer primes.

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Jiuzhou Zhao, Ruofan Li. 2025-12-04. On $β$-adic expansions of powers of algebraic integer omitting a digit. https://arxiv.org/abs/2405.06220

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