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Folds! (English) Zbl 0493.10001
In this series of three papers (see Zbl 0493.10002, Zbl 0493.10003) the authors survey a varied collection of topics which are all related to the so-called paper-folding sequences. Such sequences arise from repeatedly folding a sheet of paper, unfolding it again and considering the sequence of “upward” and “downward” bends. They have a number of highly interesting properties. For example, plane-filling curves can be constructed from them. They can also be used to construct sequences of integers $$u(h)$$ satisfying $\sup_{0\leq\theta\leq 2\pi}|\sum_0^{n-1} (1)^{u(h)}e^{ih\theta}|\leq (2+\sqrt 2)\sqrt n,$ the lower bound $$\sqrt n$$ being trivial. Some alternative ways of generating related sequences are generation by automatons and by symmetry operations. For example, $$\sum g_hX^h$$ is algebraic over $$\mathbb F_p[X]$$ if and only if the sequence $$g_h$$ can be generated by a so-called $$p$$-automaton. Moreover, $$\sum g_hp^{-h}$$ is a transcendental number in that case. Furthermore, the continued fraction of the Fredholm series $$g^{-2^h}$$ can be given by a sequence generated by symmetry operations.
By generalization of the paperfolding idea, one can construct bizarre, plane-filling curves, which, drawn on a piece of paper yield intricate patterns that arouse ones fantasy. In all, the paper contains much information, is written in an entertaining form and worth wile reading.
Reviewer: F. Beukers
Show Scanned Page ##### MSC:
 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 68Q45 Formal languages and automata 11J81 Transcendence (general theory) 11A55 Continued fractions
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