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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)
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References:

[1] R. Ayoub,Introduction to the Analytic Theory of Numbers (Math. Surveys No. 10, AMS, Rhode-Island, 1963). · Zbl 0128.04303
[2] H. Bai-Lin,Elementary Symbolic Dynamics and Chaos in Dissipative Systems (World Scientific, Singapore, 1989). · Zbl 0724.58001
[3] H.M. Edwards,Riemann’s Zeta Function (Academic Press, New York, 1974). · Zbl 0315.10035
[4] G.H. Hardy and E.M. Wright,An Introduction to the Theory of Numbers, Fourth Edition (Oxford at the Clarendon Press, London, 1971). · Zbl 0020.29201
[5] M.B. Katz,Questions of Uniqueness and Resolution in Reconstruction from Projections (Springer-Verlag, Berlin, 1978). · Zbl 0385.92002
[6] W.J. LeVeque,Topics in Number Theory, Third printing (Addison-Wesley, New-York, 1965).
[7] B.B. Mandelbrot,The Fractal Geometry of Nature (W.H. Freeman, San Francisco, 1982). · Zbl 0504.28001
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