A total order in \((0,1]\) defined through a ``next'' operator
Any real number \(\alpha\in (0,1]\) has a representation of the form (*) \(\alpha= \sum_{k=1}^\infty (-1)^{k-1} \frac{1}{a_1 a_2\dots a_k}\), where \(\{a_j\}\) is a strictly increasing sequence of natural numbers; \(\alpha\) is rational iff the series on the right-hand side of (*) is finite. If we require that in the finite case the last two \(a\)'s are not consecutive, the representation is unique. The authors establish a 1-1 correspondence between the rationals \(\alpha= \frac{1}{a_1}- \frac{1}{a_1a_2}\pm \dots\pm \frac{1}{a_1a_2\dots a_k}\), which are denoted by \((a_1a_2\dots a_k)\) and the natural numbers \(m\) having the dyadic expansion \(m= \sum_{j=1}^k 2^{k_j}\). Furthermore, they introduce the ``next'' operator \(\sigma ((a_1 a_2\dots a_k))= (1,a_1\dots a_k)\) if \(a_1> 1\) and \(\sigma ((a_1 a_2\dots a_k))= (r+1, a_r\dots a_k)\) if \(a_1= 1, a_2= 2,\dots, a_r= r\) but \(a_{r+1}> r+1\). In this way an order among the rationals is established, which is extended to the real numbers in \((0,1)\). Finally, the asymptotic distribution function of the elements \(\sigma(\alpha), \sigma^2(\alpha),\dots\) is studied, where \(\alpha= (a_1 a_2\dots)\). It turns out that this function is a continuous, strictly increasing singular function.
scientific article; zbMATH DE number 1501334