Powers of rationals modulo 1 and rational base number systems

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 letter-to-letter 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 so-called “Josephus problem”. More important, these expansions in the base p q allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.