Komornik–Loreti constant

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

Definition[edit]

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.[1]

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.[2]

References[edit]

  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 2010-12-27.