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The Rudin-Shapiro sequence, Ising chain, and paperfolding. (English) Zbl 0724.11010
Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 367-382 (1990).

[For the entire collection see Zbl 0711.00008.]

This paper is a survey of previous papers [the reviewer and the author, Bull. Soc. Math. Fr. 113, 273-283 (1985)), Mathematika 32, 33-38 (1985; Zbl 0561.10025), the author, Number Theory and Physics, Springer Proc. Phys. 47, 195-202 (1990); the author and G. Tenenbaum, Bull. Soc. Math. Fr. 109, 207-215 (1981; Zbl 0468.10033), relating paper-folding, the Rudin-Shapiro sequence and the 1D Ising chain at imaginary temperature.

If one wants to know why “Number Theory is imaginary temperature physics” or why the dragon curve with angle $\alpha$ boils for $\alpha =90$ degrees (!), one should definitely read this survey.

Note that reference [14] has appeared (see above), that the optimal constant for the Rudin-Shapiro sequence (see p. 369) has recently been obtained by B. Saffari $\left(C=\sqrt{6}\right)$ and that the theorem on page 375 has been generalized by the reviewer and P. Liardet (Generalized Rudin-Shapiro sequences, to appear in Acta Arith.)

MSC:
 11B85 Automata sequences 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 11L03 Trigonometric and exponential sums, general