Extremal values of semi-regular continuants and codings of interval exchange transformations

Abstract: 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.