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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

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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