Prouhet–Thue–Morse constant

In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet (fr), Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,

[math]\displaystyle{ \tau = \sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}} = 0.412454033640 \ldots }[/math]

where t_n is the n^th element of the Prouhet–Thue–Morse sequence.

Other representations

The Prouhet–Thue–Morse constant can also be expressed, without using t_n , as an infinite product,

[math]\displaystyle{ \tau = \frac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right)\right] }[/math]

This formula is obtained by substituting x = 1/2 into generating series for t_n

[math]\displaystyle{ F(x) = \sum_{n=0}^{\infty} (-1)^{t_n} x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ) }[/math]

The continued fraction expansion of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] (sequence A014572 in the OEIS)

Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.

Transcendence

The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.

He also showed that the number

[math]\displaystyle{ \sum_{i=0}^{\infty} t_n \, \alpha^n }[/math]

is also transcendental for any algebraic number α, where 0 < |α| < 1.

Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2.

Appearances

The Prouhet–Thue–Morse constant appears in probability. If a language L over {0, 1} is chosen at random, by flipping a fair coin to decide whether each word w is in L, the probability that it contains at least one word for each possible length is

[math]\displaystyle{ p = \prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right) = \sum_{n=0}^{\infty} \frac{(-1)^{t_n}}{2^{n+1}} = 2 - 4 \tau = 0.35018386544\ldots }[/math]

See also

• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Komornik–Loreti constant

References

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. .
• Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. 

External links

• OEIS sequence A010060 (Thue-Morse sequence)
• The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
• PlanetMath entry