Minimal redundant digit expansions in the gaussian integers
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 517-528

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.

Un résultat récent établit qu’il suffit de connaître les deux derniers chiffres significatifs du développement en base q usuel d’un entier pour calculer le dernier chiffre significatif dans le développement en base q redondant minimal. Nous montrons que l’énoncé analogue pour les entiers de Gauss est faux.

@article{JTNB_2002__14_2_517_0,
     author = {Heuberger, Clemens},
     title = {Minimal redundant digit expansions in the gaussian integers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {517-528},
     zbl = {1076.11005},
     mrnumber = {2040691},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_517_0}
}
Heuberger, Clemens. Minimal redundant digit expansions in the gaussian integers. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 517-528. http://www.numdam.org/item/JTNB_2002__14_2_517_0/

[1] C. Heuberger, H. Prodinger, On minimal expansions in redundant number systems: Algorithms and quantitative analysis. Computing 66 (2001), 377-393. | MR 1842756 | Zbl 1030.11003

[2] I. Kátai, J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. | MR 389759 | Zbl 0309.12001

[3] D.E. Knuth, Seminumerical algorithms, third ed. The Art of Computer Programming, vol. 2, Addison-Wesley, 1998. | MR 633878 | Zbl 0895.65001

[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. | MR 1152592 | Zbl 0760.11002