Geometrical aspects of expansions in complex bases

TL;DR

This paper explores the geometric structure of numbers representable in complex bases with arbitrary digits, providing descriptions of their convex hulls, extremal points, and conditions for convexity.

Contribution

It offers a novel geometric analysis of representable numbers in complex bases, including convex hull characterization and extremal points.

Findings

Explicit description of convex hulls of representable numbers

Characterization of extremal points in the convex set

Conditions for convexity of the representable set

Abstract

We study the set of the representable numbers in base $q=pe^{i\frac{2\pi}{n}}$ with $\rho>1$ 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.