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- W4386875669 abstract "Let $p$ be a prime number and $K$ be a field with embeddings into $mathbb{R}$ and $mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $mathbb{Q}_p$ and is expected to simultaneously converge in both $mathbb{R}$ and $mathbb{Q}_p$. This algorithm produces finite continued fraction expansions for rational numbers. In the case of $p=2$ and if $K$ is a quadratic field, the continued fraction expansions generated by this algorithm converge in $mathbb{R}$, and they are eventually periodic or finite. For an element $alpha$ in $K$, let $p_n/q_n$ denote the $n$-th convergent. There exist constants $u_1$ and $u_2$ in ${mathbb R}_{>0}$ with $u_1 + u_2 = 2$, and constants $C_1$ and $C_2$ in ${mathbb R}_{>0}$ such that $|alpha - p_n/q_n| < C_1/|q_n|^{u_1}$ and $|alpha - p_n/q_n|_2 < C_2/|q_n|^{u_2}$. Here, $|cdot|_2$ represents the $2$-adic distance. For prime numbers $p > 2$, we present numerical experiences." @default.
- W4386875669 created "2023-09-20" @default.
- W4386875669 creator A5074996993 @default.
- W4386875669 date "2023-09-17" @default.
- W4386875669 modified "2023-09-27" @default.
- W4386875669 title "Simultaneous Convergent Continued Fraction Algorithm for Real and $p$-adic Fields with Applications to Quadratic Fields" @default.
- W4386875669 doi "https://doi.org/10.48550/arxiv.2309.09447" @default.
- W4386875669 hasPublicationYear "2023" @default.
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