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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==
Given a real number ''q''&nbsp;>&nbsp;1, the series
 
: <math>x = \sum_{n=0}^\infty a_n q^{-n}</math>
 
is called the ''q''-expansion, or [[Non-integer representation|<math>\beta</math>-expansion]], of the positive real number ''x'' if, for all <math>n \ge 0</math>, <math>0 \le a_n \le \lfloor q \rfloor</math>, where <math>\lfloor q \rfloor</math> is the [[Floor and ceiling functions|floor function]] and <math>a_n</math> need not be an integer. Any real number <math>x</math> such that <math>0 \le x \le q \lfloor q \rfloor /(q-1)</math> has such an expansion, as can be found using the [[greedy algorithm]].
 
The special case of <math>x = 1</math>, <math>a_0 = 0</math>, and <math>a_n = 0</math> or 1 is sometimes called a <math>q</math>-development. <math>a_n = 1</math> gives the only 2-development. However, for almost all <math>1 < q < 2</math>, there are an infinite number of different <math>q</math>-developments. Even more surprisingly though, there exist exceptional <math>q \in (1,2)</math> for which there exists only a single <math>q</math>-development. Furthermore, there is a smallest number <math>1 < q < 2</math> known as the Komornik–Loreti constant for which there exists a unique <math>q</math>-development.<ref name="MW">Weissman, Eric W. "q-expansion" From [http://mathworld.wolfram.com/q-Expansion.html Wolfram MathWorld]. Retrieved on 2009-10-18.</ref>
 
The Komornik–Loreti constant is the value <math>q</math> such that
 
: <math>1 = \sum_{n=1}^\infty \frac{t_k}{q^k}</math>
 
where <math>t_k</math> is the [[Thue–Morse sequence]], i.e., <math>t_k</math> is the parity of the number of 1's in the binary representation of <math>k</math>. It has approximate value
 
: <math>q=1.787231650\ldots. \,</math>
 
The constant <math>q</math> is also the unique positive real root of
 
: <math>\prod_{k=0}^\infty \left ( 1 - \frac{1}{q^{2^k}} \right ) = \left ( 1 - \frac{1}{q} \right )^{-1} - 2.</math>
 
This constant is [[transcendental number|transcendental]].<ref name="MW2">Weissman, Eric W. "Komornik–Loreti Constant." From [http://mathworld.wolfram.com/Komornik-LoretiConstant.html Wolfram MathWorld]. Retrieved on 2010-12-27.</ref>
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Komornik-Loreti constant}}
[[Category:Mathematical constants]]

Latest revision as of 20:16, 17 September 2013

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

x=n=0anqn

is called the q-expansion, or β-expansion, of the positive real number x if, for all n0, 0anq, where q is the floor function and an need not be an integer. Any real number x such that 0xqq/(q1) 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 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.[1]

The Komornik–Loreti constant is the value q such that

1=n=1tkqk

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

q=1.787231650.

The constant q is also the unique positive real root of

k=0(11q2k)=(11q)12.

This constant is transcendental.[2]

References

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  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.