On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$
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| Format: | Preprint |
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2025
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| _version_ | 1866909545670901760 |
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| author | Hitoshi, Nakada Rie, Natsui Mako, Toyosumi |
| author_facet | Hitoshi, Nakada Rie, Natsui Mako, Toyosumi |
| contents | We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = η\cdot\mathfrak o(\sqrt{-3})$, $η= \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ and the existence of the absolutely continuous ergodic invariant probability measure for $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$ Hitoshi, Nakada Rie, Natsui Mako, Toyosumi Number Theory 11J25, 11J70, 11K50 We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = η\cdot\mathfrak o(\sqrt{-3})$, $η= \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ and the existence of the absolutely continuous ergodic invariant probability measure for $T$. |
| title | On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$ |
| topic | Number Theory 11J25, 11J70, 11K50 |
| url | https://arxiv.org/abs/2503.16077 |