Powers of rationals modulo 1 and rational base number systems
type: Item
instance of: scholarly article
title: Powers of rationals modulo 1 and rational base number systems
zbMATH Open document ID: 1214.11089
author: Shigeki Akiyama
author: Christiane Frougny
author: Jacques Sakarovitch
published in: Israel Journal of Mathematics
publication date: 2009-02-17T00:00:00Z
review text: 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 \[ \left\{z\left(\frac32\right)^n\right\}\in\left[0,\frac13\right[\cup\left[\frac23,1\right[ \quad\text{ for }n=0,1,2,\dots \]
reviewed by: Manfred G. Madritsch
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 68Q70
zbMATH DE Number: 5508715
zbMATH Keywords: radix expansion
zbMATH Keywords: powers of rationals
zbMATH Keywords: Mahler's Z-numbers
zbMATH Keywords: Josephus problem
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: s11856-008-1056-4
OpenAlex ID: W2124301002
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DOI: 10.1007/S11856-008-1056-4
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description: scientific article; zbMATH DE number 5508715