Simultaneous Convergent Continued Fraction Algorithm for Real and $p$-adic Fields with Applications to Quadratic Fields

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 $p_n\mathrm{/}{q}_n$ 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 $\mathrm{\mid }\alpha -p_n\mathrm{/}{q}_n\mathrm{\mid }<C_1\mathrm{/}\mathrm{\mid }{q}_n{\mathrm{\mid }}^{u_1}$ and $\mathrm{\mid }\alpha -p_n\mathrm{/}{q}_n{\mathrm{\mid }}_2<C_2\mathrm{/}\mathrm{\mid }{q}_n{\mathrm{\mid }}^{u_2}$. Here, $\mathrm{\mid }\cdot {\mathrm{\mid }}_2$ represents the $2$-adic distance. For prime numbers $p>2$, we present numerical experiences.