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 $n\geq 0$, $0\leq a_{n}\leq \lfloor q\rfloor$, where $\lfloor q\rfloor$ is the floor function and $a_{n}$ need not be an integer. Any real number $x$ such that $0\leq x\leq q\lfloor q\rfloor /(q-1)$ 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 the equation

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

This constant is transcendental.

See also

• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Prouhet–Thue–Morse constant