On the interval maps associated to the \(\alpha\)-mediant convergents (Q1890459)

The paper considers continued fractions of the form \[ x=a_0+ \frac{a_i|} {| b_1}+\frac{a_2|}{| b_2}+\dots, \] where \(x\) is irrational and the \(b_k'\)s are \(b_k=[(T^k_\alpha(x))^{-1}]\), \(T_\alpha(x)\) being the transformation \[ T_\alpha(x)=\left|\frac 1x\right|-\left[\left|\frac 1x\right| \right]_\alpha;\tag{*} \] here \([y]_\alpha=n\) for \(y\in[n-1+\alpha,n+\alpha)\). \(\alpha=1\) gives us the regular continued fraction expansion of \(x\), \(\alpha= 1/2\) the continued fraction expansion to the nearest integer. The paper deals with the convergence to \(x\) by the convergents of the and also with the behaviour of the medians. A similar investigation is carried out about a transformation \(F(x)=x/(1-x)\) for \(x\in[0,1/2)\), \(F(x)=(1-x)/x\) for \(x\in [1/2,1)\). Reviewer's remark: A closed expression for \(\{kx\}\), \((\{\cdot\}\) is the fractional part) which contains also the case when \(\{kx\}\) is ``small'' can be obtained by using the Ostrowski algorithm described on pp. 23--27 of the book ``Continued Fractions'' [World Scientific, Singapore (1992; Zbl 0925.11038)], written jointly by \textit{A. Rockett} and the reviewer.