The Komornik–Loreti constant is a mathematical constant that represents the smallest number for which there still exists a unique q-development.
Given a real number q > 1, the series
x = ∑ n = 0 ∞ a n q − n is called the q-expansion, or β -expansion, of the positive real number x if, for all n ≥ 0 , 0 ≤ a n ≤ ⌊ q ⌋ , where ⌊ q ⌋ is the floor function and a n need not be an integer. Any real number x such that 0 ≤ x ≤ q ⌊ q ⌋ / ( q − 1 ) 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 ∈ ( 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.
The Komornik–Loreti constant is the value q such that
1 = ∑ n = 1 ∞ t k q k 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
q = 1.787231650 … . The constant q is also the unique positive real root of
∏ k = 0 ∞ ( 1 − 1 q 2 k ) = ( 1 − 1 q ) − 1 − 2. This constant is transcendental.
