Komornik–Loreti constant explained

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

x = \sum ^{inf ty} _{n =0} a _{n} q ^{-n}

is called the q-expansion, or

\beta -expansion, of the positive real number x if, for all

n \ge 0 , 0 \le a _{n} \le \lfloor q \rfloor , where \lfloor q \rfloor is the floor function and a _{n} need not be an integer. Any real number x such that 0 \le x \le q \lfloor q \rfloor /(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 \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.^[1]

Value

The Komornik–Loreti constant is the value

q such that1 = \sum ^{inf ty} _{k =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 valueq =1 . 787231650 \ldots. ^[2]

The constant

q is also the unique positive real solution to\prod ^{inf ty} _{k =0} \left ( 1 - (1)/(q ^{2 ^{k}}) \right ) = \left ( 1 - (1)/(q) \right ) ^{-1} - 2 .

This constant is transcendental.

See also

• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Prouhet–Thue–Morse constant

Notes and 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 2010-12-27.