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Auto-similarity in Rational Base Number Systems

  • Shigeki Akiyama
  • Victor Marsault
  • Jacques Sakarovitch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8079)

Abstract

This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the integers and whose subtrees are all distinct. With every node of that tree is then associated a minimal infinite word (and a maximal infinite word).

The main result is that a sequential transducer which computes for all n the minimal word associated with n + 1 from the one associated with n, has essentially the same underlying graph as the tree itself.

These infinite words are then interpreted as representations of real numbers; the difference between the numbers represented by the maximal and minimal word associated with n is called the span of n. The preceding construction allows to characterise the topological closure of the set of spans.

Keywords

Number System Free Monoid Topological Closure Infinite Word Integer Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shigeki Akiyama
    • 1
  • Victor Marsault
    • 2
  • Jacques Sakarovitch
    • 2
  1. 1.University of TsukubaTsukubaJapan
  2. 2.Telecom-ParisTech and CNRSParisFrance

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