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
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 1 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.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
The constant is also the unique positive real root of
This constant is transcendental.
