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 = ∑ n = 0 ∞ a n q − n {\displaystyle x=\sum _{n=0}^{\infty }a_{n}q^{-n}}
is called the q-expansion, or β {\displaystyle \beta } -expansion, of the positive real number x if, for all n ≥ 0 {\displaystyle n\geq 0} , 0 ≤ a n ≤ ⌊ q ⌋ {\displaystyle 0\leq a_{n}\leq \lfloor q\rfloor } , where ⌊ q ⌋ {\displaystyle \lfloor q\rfloor } is the floor function and a n {\displaystyle a_{n}} need not be an integer. Any real number x {\displaystyle x} such that 0 ≤ x ≤ q ⌊ q ⌋ / ( q − 1 ) {\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 x = 1 {\displaystyle x=1} , a 0 = 0 {\displaystyle a_{0}=0} , and a n = 0 {\displaystyle a_{n}=0} or 1 {\displaystyle 1} is sometimes called a q {\displaystyle q} -development. a n = 1 {\displaystyle a_{n}=1} gives the only 2-development. However, for almost all 1 < q < 2 {\displaystyle 1<q<2} , there are an infinite number of different q {\displaystyle q} -developments. Even more surprisingly though, there exist exceptional q ∈ ( 1 , 2 ) {\displaystyle q\in (1,2)} for which there exists only a single q {\displaystyle q} -development. Furthermore, there is a smallest number 1 < q < 2 {\displaystyle 1<q<2} known as the Komornik–Loreti constant for which there exists a unique q {\displaystyle q} -development.
Value The Komornik–Loreti constant is the value q {\displaystyle q} such that
1 = ∑ k = 1 ∞ t k q k {\displaystyle 1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}}
where t k {\displaystyle t_{k}} is the Thue–Morse sequence, i.e., t k {\displaystyle t_{k}} is the parity of the number of 1's in the binary representation of k {\displaystyle k} . It has approximate value
q = 1.787231650 … . {\displaystyle q=1.787231650\ldots .\,}
The constant q {\displaystyle q} is also the unique positive real solution to the equation
∏ k = 0 ∞ ( 1 − 1 q 2 k ) = ( 1 − 1 q ) − 1 − 2. {\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.
See also Euler–Mascheroni constant Fibonacci word Golay–Rudin–Shapiro sequence Prouhet–Thue–Morse constant
References