On simultaneous Diophantine approximation to periodic points related to modified Jacobi-Perron algorithm

Abstract

For each $(\alpha, \beta)$ which is a periodic point related to modified Jacobi-Perron algorithm and $\mathbb{Q}(\alpha)$ has a complex embedding, we claim the following facts: the limit set of $\{(\sqrt{q_n} (q_n \alpha - p_n), \sqrt{q_n} (q_n \beta - r_n) | n = 1, 2, \dots\}$ is a finite union of similar ellipses, where $(p_n, q_n, r_n)$ is the $n$-th convergent $(p_n/q_n, r_n/q_n)$ of $(\alpha, \beta)$ by the modified Jacobi-Perron algorithm but for some $(\alpha, \beta)$ the ellipse given above is not the nearest ellipse in the limit set of $\{(\sqrt{q} (q\alpha - p), \sqrt{q} (q\beta - r) | q \in \mathbb{Z}, q \gt 0\}$ which is a union of similar ellipses.