 
Summary: Powers of rationals modulo 1 and
rational base number systems
Shigeki Akiyama
Christiane Frougny
Jacques Sakarovitch
6 April 2006
Abstract
A new method for representing positive integers and real numbers in a rational
base is considered. It amounts to computing the digits from right to left, least
significant first. Every integer has a unique such expansion. The set of expansions
of the integers is not a regular language but nevertheless addition can be performed
by a lettertoletter finite right transducer. Every real number has at least one such
expansion and a countable infinite set of them have more than one. We explain how
these expansions can be approximated and characterize the expansions of reals that
have two expansions.
These results are not only developed for their own sake but also as they relate
to other problems in combinatorics and number theory. A first example is a new
interpretation and expansion of the constant K(p) from the socalled "Josephus
problem". More important, these expansions in the base p
q allow us to make some
