Geometrical aspects of expansions in complex bases

We study the set of the representable numbers in base $q=pe^{i\frac{2\pi}{n}}$ with ρ>1 and n∈ℕ and with digits in an arbitrary finite real alphabet A. We give a geometrical description of the convex hull of the representable numbers in base q and alphabet A and an explicit characterization of its extremal points. A characterizing condition for the convexity of the set of representable numbers is also shown.

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