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
- [math]\displaystyle{ x = \sum_{n=0}^\infty a_n q^{-n} }[/math]
is called the q-expansion, or [math]\displaystyle{ \beta }[/math]-expansion, of the positive real number x if, for all [math]\displaystyle{ n \ge 0 }[/math], [math]\displaystyle{ 0 \le a_n \le \lfloor q \rfloor }[/math], where [math]\displaystyle{ \lfloor q \rfloor }[/math] is the floor function and [math]\displaystyle{ a_n }[/math] need not be an integer. Any real number [math]\displaystyle{ x }[/math] such that [math]\displaystyle{ 0 \le x \le q \lfloor q \rfloor /(q-1) }[/math] has such an expansion, as can be found using the greedy algorithm.
The special case of [math]\displaystyle{ x = 1 }[/math], [math]\displaystyle{ a_0 = 0 }[/math], and [math]\displaystyle{ a_n = 0 }[/math] or [math]\displaystyle{ 1 }[/math] is sometimes called a [math]\displaystyle{ q }[/math]-development. [math]\displaystyle{ a_n = 1 }[/math] gives the only 2-development. However, for almost all [math]\displaystyle{ 1 \lt q \lt 2 }[/math], there are an infinite number of different [math]\displaystyle{ q }[/math]-developments. Even more surprisingly though, there exist exceptional [math]\displaystyle{ q \in (1,2) }[/math] for which there exists only a single [math]\displaystyle{ q }[/math]-development. Furthermore, there is a smallest number [math]\displaystyle{ 1 \lt q \lt 2 }[/math] known as the Komornik–Loreti constant for which there exists a unique [math]\displaystyle{ q }[/math]-development.[2]
Value
The Komornik–Loreti constant is the value [math]\displaystyle{ q }[/math] such that
- [math]\displaystyle{ 1 = \sum_{k=1}^\infty \frac{t_k}{q^k} }[/math]
where [math]\displaystyle{ t_k }[/math] is the Thue–Morse sequence, i.e., [math]\displaystyle{ t_k }[/math] is the parity of the number of 1's in the binary representation of [math]\displaystyle{ k }[/math]. It has approximate value
- [math]\displaystyle{ q=1.787231650\ldots. \, }[/math][3]
The constant [math]\displaystyle{ q }[/math] is also the unique positive real root of
- [math]\displaystyle{ \prod_{k=0}^\infty \left ( 1 - \frac{1}{q^{2^k}} \right ) = \left ( 1 - \frac{1}{q} \right )^{-1} - 2. }[/math]
This constant is transcendental.[4]
See also
- Euler-Mascheroni constant
- Fibonacci word
- Golay–Rudin–Shapiro sequence
- Prouhet–Thue–Morse constant
References
- ↑ Komornik, Vilmos; Loreti, Paola (1998), "Unique developments in non-integer bases", American Mathematical Monthly 105 (7): 636–639, doi:10.2307/2589246
- ↑ Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
- ↑ Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.
- ↑ Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly 107 (5): 448–449, doi:10.2307/2695302
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