Powers of rationals modulo 1 and rational base number systems
Let \(p>q\geq1\) be coprime integers. In the present paper the authors consider expansions of positive integers \(N\) in \((p/q)\)-expansion, \textit{i.e.} expansions of the form \[ N=\sum_{k=0}^\ell\frac{a_k}{q}\left(\frac pq\right)^k \] where \(a_k\in\{0,\dots, p-1\}\). They are able to show that every positive integer \(N\) has a unique and finite \((p/q)\)-expansion, by recursively calculating \(N_k\) (\(k\geq0\)) with \[ N_0=N\quad\text{and}\quad pN_k=qN_{k+1}+a_k. \] Since this expansion is calculated in a right to left manner starting from the least significant digit, it is not so obvious, how to calculate the expansion of real numbers. However, they provide an algorithm that calculates the expansion in a left to right manner for a fixed precision. Nevertheless the expansion of a real number need not be unique. A further point of investigation is the language of digits that is generated by this expansion. As the authors show, this language is neither regular nor context-free Finally they provide some applications in number theory (namely Mahler's \(Z\)-numbers and the Josephus problem) which motivate further consideration of this number system. One of their results in this direction considers the distribution of \(\left(z\left(\frac32\right)^n\right)_{n\geq0}\) modulo 1. In particular, they were able to show that there are countable infinitely many \(z\) such that \[ \left\{z\left(\frac32\right)^n\right\}\in\left[0,\frac13\right[\cup\left[\frac23,1\right[ \quad\text{ for }n=0,1,2,\dots \]
scientific article; zbMATH DE number 5508715
scientific article