Let 
be an irrational number, define 
, and let 
be the sequence obtained by arranging
the elements of 
in increasing order. A sequence 
is said to be a signature sequence if there exists
a positive irrational
number 
such that 
,
and 
is called the signature of 
.
One can also define two extended signature sequences for positive rational 
by taking the 
in increasing order or decreasing order. These can be considered
signature sequences for 
and 
, respectively, where 
is an infinitesimal.
The signature of an irrational number or either signature of a rational number is a fractal sequence.
Also, if 
is a signature or extended signature sequence, then the lower-trimmed
subsequence is 
.
It has been conjectured that every sequence with both of these properties is a signature
or extended signature sequence.
If every initial subsequence of a sequence 
is an initial subsequence of some signature sequence, then
is either a signature sequence, an extended signature sequence, or one of the two
limiting cases: all 1's, or the natural numbers (which could be regarded as signature
sequences for zero and infinity).
