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On the irrationality exponent of the regular paperfolding numbers. (English) Zbl 1368.11068
Summary: In this paper, improving the method of J.-P. Allouche et al. [Ann. Inst. Fourier 48, No. 1, 1–27 (1998; Zbl 0974.11010)], we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence modulo 2 is periodic with period 10 which answers M. Coons’s conjecture [Ramanujan J. 30, No. 1, 39–65 (2013; Zbl 1271.11075)]. Then we extend Bugeaud’s method to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.

MSC:
11J82 Measures of irrationality and of transcendence
11B85 Automata sequences
11C20 Matrices, determinants in number theory
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