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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 \((C=\sqrt{6})\) 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 arising in equilibrium statistical mechanics
11L03 Trigonometric and exponential sums (general theory)