Convex hull of powers of a complex number, trinomial equations and the Farey sequence. (English) Zbl 0758.11034
For a complex number \(z\) in the open unit disk \(D\) let \(C(z)\) denote the convex hull of the sequence of integral powers \(1,z,z^ 2,\dots\) . It is proved that \(C(z)\) is a polygon (if \(z\) is not a positive real number). If \(n=n(z)\) is the number of vertices of this polygon (the so called “color” of \(z\)), then the vertices of \(C(z)\) are precisely the first \(n\) powers of \(z\). This induces a coloring of \(D\). Furthermore the boundary of the coloring (i.e. the points where the coloring changes) is studied. If \(z\) is a boundary point with color \(n\), then there exists a positive integer \(k<n\) and \(\alpha\in(0,1)\) such that \(z^ n=\alpha z^ k+1-\alpha\). Such trinomial equations are investigated in detail. This yields a precise description of the coloring structure: the boundary points form a fractal set which is the countable union of rectifiable curves, thus the Hausdorff dimension is 1. Finally, three algorithms for computing the coloring are established and some problems and connections to Farey numbers are discussed.
Reviewer: R.F.Tichy (Graz)
MSC:
11K06 | General theory of distribution modulo \(1\) |
52A10 | Convex sets in \(2\) dimensions (including convex curves) |
11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
Keywords:
convex sets; geometric algorithms; color of complex number; number of vertices of the convex hull of powers of a complex number; coloring of the unit disk; fractalsReferences:
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