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Powers of rationals modulo 1 and rational base number systems

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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 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 number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.

The results that we derive are pertinent on their own and 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 \( \tfrac{p} {q} \) allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Niigata University, Niigata, Japan

    Shigeki Akiyama

  2. LIAFA, UMR 7089 CNRS, Paris, France

    Christiane Frougny

  3. Université Paris 8, Paris, France

    Christiane Frougny

  4. LTCI, UMR 5141, CNRS / ENST, 46, rue Barrault, 75634, Paris Cedex 13, France

    Jacques Sakarovitch

Authors
  1. Shigeki Akiyama
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  2. Christiane Frougny
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  3. Jacques Sakarovitch
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Corresponding author

Correspondence to Shigeki Akiyama.

Additional information

Work partially supported by the CNRS/JSPS contract 13 569, and by the “ACI Nouvelles Interfaces des Mathématiques”, contract 04 312.

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Akiyama, S., Frougny, C. & Sakarovitch, J. Powers of rationals modulo 1 and rational base number systems. Isr. J. Math. 168, 53–91 (2008). https://doi.org/10.1007/s11856-008-1056-4

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  • DOI: https://doi.org/10.1007/s11856-008-1056-4

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