Extremal values of continuants

Abstract

The following question was posed by C. A. Nicol: Given an arbitrary set B B of positive integers, find the extremal denominators of regular continued fractions with partial denominators from B B , each element occurring a given number of times. Partial solutions have been given by T. S. Motzkin and E. G. Straus, and later by T. W. Cusick. We derive the general solutions from a purely combinatorial theorem about the set of permutations of a vector with components from an arbitrary linearly ordered set. We also consider certain halfregular continued fractions. Here the maximizing arrangements have to be described in terms of an algorithmic procedure, as their combinatorial structure is exceptionally complicated. Its investigation leads to a connection with the well-known Markov spectrum. Finally we obtain an asymptotic formula for the ratio of extremal continuants and some sharp (essentially analytic) inequalities concerning cyclic continuants.