On a problem of Gauss-Kuzmin type for continued fraction with odd partial quotients. (English) Zbl 0563.28012

Kalpazidou, Sofia

Pac. J. Math. 123, 103-114 (1986).

Let x be a number of the unit interval. Then x may be written in a unique way as a continued fraction \(x=1/(\alpha_ 1(x)+\epsilon_ 1(x)/(\alpha_ 2(x)+\epsilon_ 2(x)/(\alpha_ 3(x)+...))\) where \(\epsilon_ n\in \{-1,1\}\), \(\alpha_ n\geq 1\), \(\alpha_ n\equiv 1(mod 2)\) and \(\alpha_ n+\epsilon_ n>1\). Using the ergodic behaviour of a certain homogeneous random system with complete connections we solve a variant of Gauss-Kuzmin problem for the above expansion.

Reviewer: F.Schweiger

MSC:

28D05	Measure-preserving transformations
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A55	Continued fractions

Keywords:

continued fraction; ergodic behaviour of a certain homogeneous random system with complete connections; Gauss-Kuzmin problem