Extremal values of continuants and transcendence of certain continued fractions
This paper studies the transcendence of real numbers having a continued fraction expansion with bounded partial quotients for which the density of occurrence of each of the values taken by the partial quotients does exist. A transcendence criterion is first given in the continuation of [\textit{J.-P. Allouche, J. L. Davison, M. Queffélec} and \textit{L. Q. Zamboni}, J. Number Theory 91, 39--66 (2001; Zbl 0998.11036)], expressed in terms of the growth rate of the denominators of the convergents. The proof mainly uses a result on extremal values of continuants by \textit{G. Ramharter} [Proc. Am. Math. Soc. 89, 189--201 (1983; Zbl 0489.10009)] (a complete and clear proof of this latter result is given), as well as a theorem by \textit{W. M. Schmidt} [Acta Math. 119, 27--50 (1967; Zbl 0173.04801)]. This criterion is then applied to the real numbers with continued fraction expansion of the form \([0;a_1,a_2,\dots]\), with \(a_n=1+([n \theta]) \mod d\), where \(\theta\) is irrational and \(d \geq 2\) is a positive integer. Some of those results have been independently obtained in [\textit{J. L. Davison}, Proc. Edinb. Math. Soc. 45, 653--671 (2002; Zbl 1107.11303)] with a different method.
scientific article; zbMATH DE number 2107262
scientific article