On Convergents of Proper Continued Fractions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909418201808896 |
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| author | Langeveld, Niels Ralston, David |
| author_facet | Langeveld, Niels Ralston, David |
| contents | Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their convergents and classify which pairs of integers $p,q$ yield a convergent $p/q$ to some irrational $x$. Notably, we reduce the problem to finding convergence only of index one and two. We completely classify the possible choices for convergents of odd index and provide a near-complete classification for even index. We furthermore propose a natural two-dimensional generalization of the classical Gauss map as a method for dynamically generating all possible expansions and establish ergodicity of this map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05077 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Convergents of Proper Continued Fractions Langeveld, Niels Ralston, David Dynamical Systems Number Theory 11J70, 37E05, 37A44 Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their convergents and classify which pairs of integers $p,q$ yield a convergent $p/q$ to some irrational $x$. Notably, we reduce the problem to finding convergence only of index one and two. We completely classify the possible choices for convergents of odd index and provide a near-complete classification for even index. We furthermore propose a natural two-dimensional generalization of the classical Gauss map as a method for dynamically generating all possible expansions and establish ergodicity of this map. |
| title | On Convergents of Proper Continued Fractions |
| topic | Dynamical Systems Number Theory 11J70, 37E05, 37A44 |
| url | https://arxiv.org/abs/2412.05077 |