Powers of rationals modulo 1 and rational base number systems

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 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.

Rational base number systems forp-adic numbers

“…In this section we will study the rational base number system proposed by S. Akiyama, C. Frougny, and J. Sakarovitch in [1]. We will show that this system is a natural generalization of the standard way of representing p-adic numbers described in the previous section.…”

“…We will show that this system is a natural generalization of the standard way of representing p-adic numbers described in the previous section. In [1] the system is proposed as a new method to represent the non-negative integers in the form of a power series in p q :…”

“…Doing this prolongation repetitively, n approaches to infinity and we can get even irrational numbers. Such infinite representations are then studied in [1] and they turn out to be very interesting and to relate to old and difficult problems of Number Theory; namely, Mahler's problem [6] and Josephus problem [9], [8].…”

“…Let us consider the following algorithm introduced in [1] and named modified division (MD) algorithm.…”

“…We will study a possibility of using rational base representation. Our starting point will be the number system introduced by Akiyama, Frougny, and Sakarovitch in [1]. They studied finite representations of the positive integers of the form of m k=0 a k q p q k , where p > q ≥ 1 are coprime integers (p may not be prime!…”