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
is called the q-expansion, or -expansion, of the positive real number x if, for all , , where is the floor function and need not be an integer. Any real number such that has such an expansion, as can be found using the greedy algorithm.
The special case of , , and or is sometimes called a -development. gives the only 2-development. However, for almost all , there are an infinite number of different -developments. Even more surprisingly though, there exist exceptional for which there exists only a single -development. Furthermore, there is a smallest number known as the Komornik–Loreti constant for which there exists a unique -development.Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
Value The Komornik–Loreti constant is the value such that
where is the Thue–Morse sequence, i.e., is the parity of the number of 1's in the binary representation of . It has approximate value
Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.
The constant is also the unique positive real solution to the equation
This constant is transcendental.
See also Euler–Mascheroni constant Fibonacci word Golay–Rudin–Shapiro sequence Prouhet–Thue–Morse constant
References