Minimal redundant digit expansions in the gaussian integers

Clemens Heuberger

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 517-528
  • ISSN: 1246-7405

Abstract

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We consider minimal redundant digit expansions in canonical number systems in the gaussian integers. In contrast to the case of rational integers, where the knowledge of the two least significant digits in the “standard” expansion suffices to calculate the least significant digit in a minimal redundant expansion, such a property does not hold in the gaussian numbers : We prove that there exist pairs of numbers whose non-redundant expansions agree arbitrarily well but which have different least significant digits in minimal redundant expansions.

How to cite

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Heuberger, Clemens. "Minimal redundant digit expansions in the gaussian integers." Journal de théorie des nombres de Bordeaux 14.2 (2002): 517-528. <http://eudml.org/doc/248905>.

@article{Heuberger2002,
abstract = {We consider minimal redundant digit expansions in canonical number systems in the gaussian integers. In contrast to the case of rational integers, where the knowledge of the two least significant digits in the “standard” expansion suffices to calculate the least significant digit in a minimal redundant expansion, such a property does not hold in the gaussian numbers : We prove that there exist pairs of numbers whose non-redundant expansions agree arbitrarily well but which have different least significant digits in minimal redundant expansions.},
author = {Heuberger, Clemens},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {minimal redundant expansions; Gaussian integers},
language = {eng},
number = {2},
pages = {517-528},
publisher = {Université Bordeaux I},
title = {Minimal redundant digit expansions in the gaussian integers},
url = {http://eudml.org/doc/248905},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Heuberger, Clemens
TI - Minimal redundant digit expansions in the gaussian integers
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 517
EP - 528
AB - We consider minimal redundant digit expansions in canonical number systems in the gaussian integers. In contrast to the case of rational integers, where the knowledge of the two least significant digits in the “standard” expansion suffices to calculate the least significant digit in a minimal redundant expansion, such a property does not hold in the gaussian numbers : We prove that there exist pairs of numbers whose non-redundant expansions agree arbitrarily well but which have different least significant digits in minimal redundant expansions.
LA - eng
KW - minimal redundant expansions; Gaussian integers
UR - http://eudml.org/doc/248905
ER -

References

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  1. [1] C. Heuberger, H. Prodinger, On minimal expansions in redundant number systems: Algorithms and quantitative analysis. Computing66 (2001), 377-393. Zbl1030.11003MR1842756
  2. [2] I. Kátai, J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. Zbl0309.12001MR389759
  3. [3] D.E. Knuth, Seminumerical algorithms, third ed. The Art of Computer Programming, vol. 2, Addison-Wesley, 1998. Zbl0895.65001MR633878
  4. [4] B. Kovács, A. Pethö, Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math.(Szeged) 55 (1991), 287-299. Zbl0760.11002MR1152592

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