Counting rationals and diophantine approximation in missing-digit Cantor sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917206772678656 |
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| author | Chow, Sam Varjú, Péter P. Yu, Han |
| author_facet | Chow, Sam Varjú, Péter P. Yu, Han |
| contents | We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution of diophantine exponents in missing-digit sets.
A key tool in our study is Fourier $\ell^1$ dimension introduced by the last named author in [H. Yu, Rational points near self-similar sets, arXiv:2101.05910]. An important technical contribution of the paper is a method to compute this quantity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_18395 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting rationals and diophantine approximation in missing-digit Cantor sets Chow, Sam Varjú, Péter P. Yu, Han Number Theory 11J83 (primary), 28A78, 28A80, 42A16 (secondary) We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution of diophantine exponents in missing-digit sets. A key tool in our study is Fourier $\ell^1$ dimension introduced by the last named author in [H. Yu, Rational points near self-similar sets, arXiv:2101.05910]. An important technical contribution of the paper is a method to compute this quantity. |
| title | Counting rationals and diophantine approximation in missing-digit Cantor sets |
| topic | Number Theory 11J83 (primary), 28A78, 28A80, 42A16 (secondary) |
| url | https://arxiv.org/abs/2402.18395 |