The Komornik–Loreti constant is a mathematical constant that represents the smallest number for which there still exists a unique q-development.
Given a real number q > 1, the series
x
=
∑
n
=
0
∞
a
n
q
−
n
is called the q-expansion, or
β
-expansion, of the positive real number x if, for all
n
≥
0
,
0
≤
a
n
≤
⌊
q
⌋
, where
⌊
q
⌋
is the floor function and
a
n
need not be an integer. Any real number
x
such that
0
≤
x
≤
q
⌊
q
⌋
/
(
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
∈
(
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.
The Komornik–Loreti constant is the value
q
such that
1
=
∑
n
=
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 value
q
=
1.787231650
…
.
The constant
q
is also the unique positive real root of
∏
k
=
0
∞
(
1
−
1
q
2
k
)
=
(
1
−
1
q
)
−
1
−
2.
This constant is transcendental.
