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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.06220 |
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| _version_ | 1866915652244078592 |
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| author | Zhao, Jiuzhou Li, Ruofan |
| author_facet | Zhao, Jiuzhou Li, Ruofan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $β$-adic expansions of powers of algebraic integer omitting a digit Zhao, Jiuzhou Li, Ruofan Number Theory 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. |
| title | On $β$-adic expansions of powers of algebraic integer omitting a digit |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.06220 |