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

Abstract

Given a set $A$ consisting of positive integers and a k-term partition $P:n_1+n_2+\cdots +n_k=n$, find the extremal denominators of the regular and semi-regular continued fraction $[0;x_1,x_2,\text{…},x_n]$ with partial quotients $x_i\in A$ and where each $a_i$ occurs precisely $n_i$ times in the sequence $x_1,x_2,\text{…},x_n$. 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 positive integers $a_i$. However, an explicit determination of a maximizing arrangement for the semi-regular continuant turned out to be substantially 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 actual values of the $a_i$. He further verified the conjecture in the special case of a binary alphabet. In this paper, we confirm Ramharter's conjecture for sets $A$ with $\left|A\right|=3$ and give an algorithmic procedure for constructing the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with $\left|A\right|⩾4$ in that 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. This combinatorial condition may also be stated more or less verbatum in the context of infinite sequences on an ordered set. We show that in the context of 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.