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,

τ = ∑ n = 0 ∞ t n 2 n + 1 = 0.412454033640 … {\displaystyle \tau =\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}=0.412454033640\ldots }

where tn is the nth element of the Prouhet–Thue–Morse sequence.

Other representations

The Prouhet–Thue–Morse constant can also be expressed, without using tn , as an infinite product,[1]

τ = 1 4 [ 2 − ∏ n = 0 ∞ ( 1 − 1 2 2 n ) ] {\displaystyle \tau ={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]}

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

F ( x ) = ∑ n = 0 ∞ ( − 1 ) t n x n = ∏ n = 0 ∞ ( 1 − x 2 n ) {\displaystyle F(x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})}

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.[2]

Transcendence

The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.[3]

He also showed that the number

∑ i = 0 ∞ t n α n {\displaystyle \sum _{i=0}^{\infty }t_{n}\,\alpha ^{n}}

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.[4]

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 [5]

p = ∏ n = 0 ∞ ( 1 − 1 2 2 n ) = ∑ n = 0 ∞ ( − 1 ) t n 2 n + 1 = 2 − 4 τ = 0.35018386544 … {\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 }

See also

Notes

  1. ^ Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
  2. ^ Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3).
  3. ^ Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929.
  4. ^ Bugaeud, Yann (2011). "On the rational approximation to the Thue–Morse–Mahler numbers". Annales de l'Institut Fourier. 61 (5): 2065–2076. doi:10.5802/aif.2666.
  5. ^ Allouche, Jean-Paul; Shallit, Jeffrey (1999). "The Ubiquitous Prouhet-Thue-Morse Sequence". Discrete Mathematics and Theoretical Computer Science: 11.

References

  • Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015..
  • Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.

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


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