Komornik–Loreti constant

The Komornik–Loreti constant is a mathematical constant that represents the smallest number for which there still exists a unique q-development.

Definition

Given a real number q > 1, the series

is called the q-expansion, of the positive real number x if, for all , , where is the floor function and an 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, a0 = 0, and an = 0 or 1 is sometimes called a q-development. an = 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 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.[1]

The Komornik–Loreti constant is the value q such that

where tk is the Thue–Morse sequence, i.e., tk 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 root of

This constant is transcendental.[2]

References

1. ^ Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
2. ^ Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2009-10-18.

Mathematical Constants