On Periodicity of Continued fractions with Partial Quotients in Quadratic Number Fields

Abstract: The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify elements with periodic continued fraction expansions, akin to Lagrange's theorem. In this paper, we fix a real quadratic field K and take an ultimately periodic continued fraction with partial quotients in $\mathcal{O}_K$. We analyze its convergence and the growth of absolute values of $\mathcal{Q}$-pairs, and establish necessary and sufficient conditions for a real quartic irrational to have an ultimately periodic continued fraction converging to it with partial quotients in $\mathcal{O}_K$. Additionally, we analyze a specific example with $K=\mathbb{Q}(\sqrt{5})$. By the obtained results, we give a continued fraction expansion algorithm for those real quartic irrationals $\xi$ belong to a quadratic extension of $K$, whose algebraic conjugates are all real. We prove that the expansion obtained from the algorithm is ultimately periodic and convergent to the specified $\xi$.