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

: $x = \sum_^\infty a_n q^$

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.Weissman, Eric W. "q-expansion" Fro
Wolfram MathWorld
Retrieved on 2009-10-18.

Value

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

: $1 = \sum_^\infty \frac$

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

: $q=1.787231650\ldots. \,$Weissman, Eric W. "Komornik–Loreti Constant." Fro
Wolfram MathWorld
Retrieved on 2010-12-27.

The constant $q$ is also the unique positive real root of

: $\prod_^\infty \left ( 1 - \frac \right ) = \left ( 1 - \frac \right )^ - 2.$

This constant is transcendental.

See also

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

References

{{DEFAULTSORT:Komornik-Loreti constant
Mathematical constants
Non-standard positional numeral systems