Given a set $A$ of positive integers $a_1<\cdots<a_k$ and a partition $P: n_1+\cdots+n_k=n$, find the extremal denominators of the regular and semi-regular continued fraction $[0;x_1,\ldots,x_n]$ with partial quotients $x_i\in A$ and where each $a_i$ occurs exactly $n_i$ times in $x_1,\ldots,x_n$. In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi-regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the integers $a_i$. However, an explicit determination of a maximizing arrangement for the semi-regular continuant turned out to be more difficult. Ramharter conjectured that as in the other three cases, the maximizing arrangement is unique up to reversal and depends only on the partition $P$ and not on the values of the $a_i$. He further verified the conjecture in the case of a binary $A$. In this paper we confirm Ramharter's conjecture for sets $A$ with $|A|=3$ and give an algorithmic construction for the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with $|A|\geq 4$, as the maximizing arrangement is in general neither unique nor independent of the values of the digits in $A$. The central idea is that the extremal arrangements satisfy a strong combinatorial condition, which may also be stated in the context of infinite sequences on an ordered set. We show that for bi-infinite binary words, this condition coincides with the Markoff property, discovered by A.A. Markoff in 1879 in his study of minima of binary quadratic forms. We further show that this same combinatorial condition is the fundamental property which describes the orbit structure of the natural codings of points under a symmetric $k$-interval exchange transformation.

翻译：给定正整数集合 $A=\{a_1<\cdots<a_k\}$ 与划分 $P: n_1+\cdots+n_k=n$，考虑部分商 $x_i\in A$ 且每个 $a_i$ 在序列 $x_1,\ldots,x_n$ 中恰好出现 $n_i$ 次的有限正则与半正则连分数 $[0;x_1,\ldots,x_n]$ 的极值分母问题。1983年，G. Ramharter 给出了正则连分数极值排列以及半正则连分数极小排列的显式刻画，并指出每种情形下的排列在反转意义下唯一且与整数 $a_i$ 的具体取值无关。然而，半正则连续分数极大排列的显式确定更为困难。Ramharter 猜想该极大排列与其他三种情形类似，在反转意义下唯一且仅依赖于划分 $P$ 而与 $a_i$ 取值无关，并验证了该猜想在 $A$ 为二元集时的正确性。本文对 $|A|=3$ 的集合 $A$ 证实了 Ramharter 猜想，并给出了唯一极大排列的算法构造。同时证明当 $|A|\geq 4$ 时该猜想不成立——极大排列在一般情况下既非唯一，也非独立于 $A$ 中数字的取值。核心思想在于极值排列满足一个强组合条件，该条件亦可表述为有序集合上无限序列的语境。我们证明对于双无限二元字，该条件等价于 A.A. Markoff 在1879年研究二元二次型最小值时发现的 Markoff 性质，并进一步揭示这一组合条件是描述对称 $k$-区间交换变换下点的自然编码轨道结构的基本性质。