On the continued fraction expansion of a class of numbers

Abstract

We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a real number to have bounded partial quotients and construct new examples of extremal real numbers with this property. These include all real numbers whose sequence of partial quotients coincide up to its first terms with the limit of a Fibonacci sequence of words on the set of positive integers, starting with two non-commuting words (so that the limit is not ultimately periodic).