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

Given a set A consisting of positive integers a1 < a2 < · · · < ak and a k-term partition P : n1 + n2 + · · · + nk = n, find the extremal denominators of the regular and semi-regular continued fraction [0; x1, x2, . . . , xn] with partial quotients xi ∈ A and where each ai occurs precisely ni times in the sequence x1, x2, . . . , xn. 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 ai. However, the determination of the maximizing arrangement for the semi-regular continuant turned out to be more difficult. He showed that if |A| = 2, then the maximizing arrangement is unique (up to reversal) and depends only on the partition P and not on the values of the ai. He further conjectured that this should be true for general A with |A| ≥ 2. In this paper, we confirm Ramharter’s conjecture for sets A with |A| = 3 and give an algorithmic procedure for constructing the maximizing arrangement. We also show that Ramharter’s conjecture fails in general for sets with |A| ≥ 4 in that the maximizing arrangement is neither unique nor independent of the values of the digits in A. The central idea, as discovered by Ramharter, is that the extremal arrangements satisfy a strong combinatorial condition. 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 show that this same combinatorial condition, in the framework of infinite words over a k-letter alphabet, is the fundamental characterizing property which describes the orbit structure of codings of points under a symmetric k-interval exchange transformation.