Komornik–Loreti constant

In the mathematical theory of non-standard positional numeral systems, the Komornik–Loreti constant is a mathematical constant that represents the smallest base q for which the number 1 has a unique representation, called its q-development. The constant is named after Vilmos Komornik and Paola Loreti, who defined it in 1998.

Definition
Given a real number q > 1, the series


 * $$x = \sum_{n=0}^\infty a_n q^{-n}$$

is called the q-expansion, or $\beta$-expansion, of the positive real number x if, for all $$n \ge 0$$, $$0 \le a_n \le \lfloor q \rfloor$$, where $$\lfloor q \rfloor$$ is the floor function and $$a_n$$ need not be an integer. Any real number $$x$$ such that $$0 \le x \le q \lfloor q \rfloor /(q-1)$$ has such an expansion, as can be found using the greedy algorithm.

The special case of $$x = 1$$, $$a_0 = 0$$, and $$a_n = 0$$ or $$1$$ is sometimes called a $$q$$-development. $$a_n = 1$$ gives the only 2-development. However, for almost all $$1 < q < 2$$, there are an infinite number of different $$q$$-developments. Even more surprisingly though, there exist exceptional $$q \in (1,2)$$ for which there exists only a single $$q$$-development. Furthermore, there is a smallest number $$1 < q < 2$$ known as the Komornik–Loreti constant for which there exists a unique $$q$$-development.

Value
The Komornik–Loreti constant is the value $$q$$ such that


 * $$1 = \sum_{k=1}^\infty \frac{t_k}{q^k}$$

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


 * $$q=1.787231650\ldots. \,$$

The constant $$q$$ is also the unique positive real solution to


 * $$\prod_{k=0}^\infty \left ( 1 - \frac{1}{q^{2^k}} \right ) = \left ( 1 - \frac{1}{q} \right )^{-1} - 2.$$

This constant is transcendental.