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.^[1]

Definition

Given a real number q > 1, the series

${\displaystyle x=\sum _{n=0}^{\infty }a_{n}q^{-n}}$

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

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

Value

The Komornik–Loreti constant is the value ${\displaystyle q}$ such that

${\displaystyle 1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}}$

where ${\displaystyle t_{k}}$ is the Thue–Morse sequence, i.e., ${\displaystyle t_{k}}$ is the parity of the number of 1's in the binary representation of ${\displaystyle k}$. It has approximate value

${\displaystyle q=1.787231650\ldots .\,}$^[3]

The constant ${\displaystyle q}$ is also the unique positive real solution to the equation

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

This constant is transcendental.^[4]

See also

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

References

1. Komornik, Vilmos; Loreti, Paola (1998), "Unique developments in non-integer bases", American Mathematical Monthly, 105 (7): 636–639, doi:10.2307/2589246, JSTOR 2589246, MR 1633077
2. Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
3. Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.
4. Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449, doi:10.2307/2695302, JSTOR 2695302, MR 1763399