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 \beta-expansion, of the positive real number x if, for all n \ge 0,, where is the floor function and a_n need not be an integer. Any real number x such that 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.

Value

The Komornik–Loreti constant is the value q such that 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 The constant q is also the unique positive real solution to the equation This constant is transcendental.