Lower rational approximations and Farey staircases
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916112355033088 |
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| author | Richman, David Harry |
| author_facet | Richman, David Harry |
| contents | For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02935 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower rational approximations and Farey staircases Richman, David Harry Number Theory 11B57 (Primary) 40A30, 11J70, 26D15, 40A25, 11B83 (Secondary) For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization. |
| title | Lower rational approximations and Farey staircases |
| topic | Number Theory 11B57 (Primary) 40A30, 11J70, 26D15, 40A25, 11B83 (Secondary) |
| url | https://arxiv.org/abs/2303.02935 |