Representations for complex numbers with integer digits
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RESEARCH
Representations for complex numbers with integer digits Paul Surer
∗
* Correspondence:
[email protected] Institut für Mathematik, Universität für Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Vienna, Austria
Abstract We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems. Keywords: Complex expansions, Radix representations Mathematics Subject Classification: 11A63, 11B85, 11R06
1 Introduction Let ζ ∈ C be non-real, fix an ε ∈ [0, 1), and define D := {−ζ μ1 + μ2 : μ1 , μ2 ∈ [−ε, 1 − ε)} ⊂ C. The set D has the shape of a parallelepiped and is a fundamental domain for the lattice Lζ generated by −ζ and 1. In Fig. 1 we see two different examples. We define the zetatransformation on D by S : D −→ D, z −→ ζ · z (mod Lζ ). In the present article we mainly concentrate on the case that |ζ | > 1 and we are interested in radix representations of (complex) numbers induced by the zeta-transformation. Indeed, provided that |ζ | > 1 we obtain for each z ∈ D by successive application of S an expansion with respect to the base ζ z= dn ζ −n with dn = ζ S n−1 (z) − S n (z) ∈ Lζ . n≥1
The setting fits into the framework of rotational beta-expansions recently introduced in [3,4] as a way to generalise the real (one-dimensional) beta-expansion to higher dimensions where the focus is set on ergodicity, soficness and invariant probability measures. Therefore, we will not discuss these topics here but refer to [3,4]. We rather concentrate on the special feature of the zeta-transformation, namely, that, due to the particular shape of our domain D, the produced digit sequences consist of integers only. Indeed, in Fig. 2 we see that integer translates of D completely cover ζ D. The actual set of digits is given
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P. Surer Res. Number Theory (2020) 6:47
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Fig. 1 The shape and the position of the fundamental domain D and the latti
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