Geometrical aspects of expansions in complex bases

We study the set of the representable numbers in base $q=pe^{i\frac{2π}{n}}$ with $ρ>1$ and $n\in \mathbb N$ and with digits in a 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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