Rational numbers in $\times b$-invariant sets
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Abstract:
Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subseteq [0,1)$ which is invariant under $\times b$ operation, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained.References
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Bibliographic Information
- Bing Li
- Affiliation: School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
- MR Author ID: 898023
- ORCID: 0000-0003-2976-5616
- Email: scbingli@scut.edu.cn
- Ruofan Li
- Affiliation: School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
- ORCID: 0000-0003-3842-176X
- Email: liruofan@scut.edu.cn
- Yufeng Wu
- Affiliation: School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
- Email: yufengwu@scut.edu.cn
- Received by editor(s): April 23, 2022
- Received by editor(s) in revised form: July 28, 2022, and September 1, 2022
- Published electronically: February 2, 2023
- Additional Notes: This research was partially supported by NSFC 11671151 and 12271176.
- Communicated by: Katrin Gelfert
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1877-1887
- MSC (2020): Primary 11A63, 37E05
- DOI: https://doi.org/10.1090/proc/16278
- MathSciNet review: 4556185