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Analysis of Alternative Digit Sets for Nonadjacent Representations

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Abstract.

It is known that every positive integer n can be represented as a finite sum of the form ∑ i a i 2i, where a i ∈ {0, 1,−1} and no two consecutive a i ’s are non-zero (“nonadjacent form”, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research.

The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1.

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Authors and Affiliations

  1. Technische Universität Graz, Austria

    Clemens Heuberger

  2. Stellenbosch University, South Africa

    Helmut Prodinger

Authors
  1. Clemens Heuberger
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  2. Helmut Prodinger
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Additional information

This paper was written while the first author was a visitor at the John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg. He thanks the centre for its hospitality. He was also supported by the grant S8307-MAT of the Austrian Science Fund.

This author is supported by the grant NRF 2053748 of the South African National Research Foundation. The research of this author was done while he was with the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg.

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Heuberger, C., Prodinger, H. Analysis of Alternative Digit Sets for Nonadjacent Representations. Mh Math 147, 219–248 (2006). https://doi.org/10.1007/s00605-005-0364-6

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  • DOI: https://doi.org/10.1007/s00605-005-0364-6

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