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, 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
- ↑ Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
- ↑ Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.