Komornik-Loreti Constant

The Komornik-Loreti constant is the value  such that

(1)

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

(2)

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

(3)

(Komornik and Loreti 1998).

The constant  is also the unique positive real root of

(4)

(Finch 2003, p. 438).

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

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."