A Gauss-Kusmin theorem for optimal continued fractions

Abstract

Abstract. A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal S-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal S-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal S-expansion.

On convergence rate in the Gauss–Kuzmin problem for θ-expansions

“…The aim of this paper is to show a two-dimensional Gauss-Kuzmin theorem for θ-expansions. Note that in the literature there are known similar results for other types of expansions (see [3,5,4,10,11]).…”

A two-dimensional Gauss--Kuzmin theorem for $N$-continued fraction expansions

“…For example, the case of the regular continued fraction (RCF) expansion was extensively studied in [4], [7] and [9]. For Hurwitz' singular continued fractions one can find the corresponding results in [5] and [14]. In [15], Sebe proved the first two-dimensional Gauss-Kuzmin theorem which leads to an estimate of the approximation error by the expansion algorithm in the grotesque continued fractions.…”

A two-dimensional Gauss-Kuzmin theorem for $N$-continued fraction expansions

“…For example, the case of the regular continued fraction (RCF) expansion was extensively studied in [2], [5] and [7]. For Hurwitz' singular continued fractions there are known the results obtained in [3] and [11]. Also, in [12] Sebe proved the first two-dimensional Gauss-Kuzmin theorem which leads to an estimate of the approximation error by the expansion algorithm in the grotesque continued fraction.…”