Chapter 4: Systems Based on Numbers

Section 5: Mathematical Constants

Nested digit sequences

The number obtained from the substitution system {1 {1, 0}, 0 {0, 1}} is approximately 0.587545966 in base 10. It is certainly conceivable that a quantity such as Feigenbaum's constant (approximately 4.6692016091) could have a digit sequence with this kind of nested structure.

From the result on page 890, the number whose digits are obtained from {1 {1, 0}, 0 {1}} is given by Sum[2^(-Floor[n GoldenRatio]), {n, }]. This number is known to be transcendental. The nth term in its continued fraction representation turns out to be 2^Fibonacci[n - 2].

The fact that nested digit sequences do not correspond to algebraic numbers follows from work by Alfred van der Poorten and others in the early 1980s. The argument is based on showing that an algebraic function always exists for which the coefficients in its power series correspond to any given nested sequence when reduced modulo some p. (See page 1092.) But then there is a general result that if a particular sequence of power series coefficients can be obtained from an algebraic (but not rational) function modulo a particular p, then it can only be obtained from transcendental functions modulo any other p—or over the integers.

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From Stephen Wolfram: A New Kind of Science [citation]