Damien Roy
Published 2004-09-14, updated 2008-05-23Version 3
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).
Comments: 16 pages, conference, small corrections in Section 4 (statement of Cor. 4.2 and proof of Prop. 4.3), addition of a new bibliographical reference
Journal: in: Diophantine approximation, Festschrift for Wolfgang Schmidt, Developments in Math. vol. 16, Eds: H.~P.~Schlickewei, K.~Schmidt and R.~Tichy, Springer-Verlag, 2008, 347--361
Transcendence and continued fraction expansion of values of Hecke-Mahler series
On the period of the continued fraction expansion of ${\sqrt {2^{2n+1}+1}}$
Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers
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