Komornik–Loreti constant

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Short description: Mathematical constant of numeral systems

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.[1]

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.[2]

Value

The Komornik–Loreti constant is the value q such that

1=k=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.[3]

The constant q is also the unique positive real root of

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

This constant is transcendental.[4]

See also

References

  1. Komornik, Vilmos; Loreti, Paola (1998), "Unique developments in non-integer bases", American Mathematical Monthly 105 (7): 636–639, doi:10.2307/2589246 
  2. Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
  3. Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.
  4. Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly 107 (5): 448–449, doi:10.2307/2695302