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Komornik-Loreti Constant

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The Komornik-Loreti constant is the value q such that

1=sum_(n=1)^infty(t_k)/(q^k),
(1)

where {t_k} is the Thue-Morse sequence, i.e., t_k is the parity of the number of 1's in the binary representation of k. It has approximate value

q=1.787231650...
(2)

(OEIS A055060). This constant is the smallest number 1<q<2 for which there is a unique q-expansion

1=sum_(i=1)^inftyepsilon_iq^(-i)
(3)

(Komornik and Loreti 1998).

The constant q is also the unique positive real root of

product_(k=0)^infty(1-1/(q^(2^k)))=(1-1/q)^(-1)-2
(4)

(Finch 2003, p. 438).

Allouche and Cosnard (2000) showed that this constant is transcendental.

See also

q-Expansion, Thue-Morse Sequence

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References

Allouche, J.-P. and Cosnard, M. "The Komornik-Loreti Constant Is Transcendental." Amer. Math. Monthly 107, 448-449, 2000.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 438-349, 2003.Komornik, V. and Loreti, P. "Unique Developments in Non-Integer Bases." Amer. Math. Monthly 105, 636-639, 1998.Sloane, N. J. A. Sequence A055060 in "The On-Line Encyclopedia of Integer Sequences."

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Komornik-Loreti Constant

Cite this as:

Weisstein, Eric W. "Komornik-Loreti Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Komornik-LoretiConstant.html

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